PRACTICAL ASTRONOMY FOR ENGINEERS SHARES PRACTICAL ASTRONOMY FOR ENGINEERS PRACTICAL ASTRONOMY FOR ENGINEERS BY FREDERICK HANLEY SEARES 111 Professor of Astronomy in the University of Missouri and Director of the Laws Observatory COLUMBIA, MISSOURI THE E. W. STEPHENS PUBLISHING COMPANY 1909 D 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 knowledge 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 a question fundaprinciples and methods of the art of numerical calculation 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 chapter. edition will contain this paper, in a revised form, as a preliminary 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 made somewhat narrowly restricted, an attempt has been means of acquiring correct and complete to place before the student the 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 consideration of the subject of time. It is not customary to introduce historical data into texts designed for the use of professional students; but the author has found so much that is PREFACE vi 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 to disregard the most effective historical of the significance and bearing a full means of giving the student appreciation is it Brief 6f scientific results. hoped that these sections though they are, material from scientific instruction is toward wider excursions into this most fascinating field. most of the examples have been printed in The both the application of the formulae involved better to illustrate detail in order to be and the operations performed by the computer. Care has been taken to may incline the reader numerical solutions for 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 of errata on page 132. use of the text should be list The For supplemented by a study of the prominent constellations. 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. this acknowledgments are due to Mr. E. S. Haynes and Mr. Harlow Shapley, of the Department of Astronomy of the University of Missouri, for My much valuable assistance in preparing the manuscript, in checking the calculations, and in reading the proofs. F. H. LAWS OBSERVATORY, UNIVERSITY OF MISSOURI, June, 1909. SEARfS. CONTENTS CHAPTER I INTRODUCTION CELESTIAL SPHERE COORDINATES , , PAGE. 1 . 2. 3. 4. 5. 6. 7. 8. 9. The results of astronomical investigations The apparent phenomena of the heavens Relation of the apparent phenomena to their 1 -' 4' '. ' interpretation Relation of the problems of practical astronomy to the phenomena of the heavens Coordinates and coordinate systems 5 7 8 Characteristics of the three systems. Changes in the coordinates Summary. Method of treating the corrections in practice Ifr Refraction Parallax 16 15 18 * CHAPTER II FORMULAE OF SPHERICAL TRIGONOMETRY TRANSFORMATION OF COORDINATES GENERAL DISCUSSION OF PROBLEMS 1 0. 1 1 . 12. 13. The fundamental formulae of spherical trigonometry Relative positions of the reference circles of the three coordinate systems Transformation of azimuth and zenith distance into hour angle and declina- 21 tion 25 '. Transformation of hour angle and declination into azimuth and zenith dis- tance 14. 15. 16. 17. 23 Transformation of hour angle into right ascension, and vice versa Transformation of azimuth and altitude into right ascension and declination, and vice versa Given the latitude of the place, and the declination and zenith distance of an object, to find its hour angle, azimuth, and parallactic angle Application of transformation formulae to the determination of latitude, azimuth, and time 29 2? 31 31 32 CHAPTER HI TIME AND TIME TRANSFORMATION 1 8. 1 9. 20. 21. basis of time measurement Apparent, or true, solar time Mean solar time Sidereal time The 23. The tropical year The calendar 24. Given the 22. 25. 26. 27. 28. 29. 30. local 36 36 36 , 37 38 ". 38 time at any point, to find the corresponding local time at any other point Given the apparent solar time at any place, to find the corresponding mean solar time, and vice versa Relation between the values of a time interval expressed in mean solar and sidereal units Relation between mean solar time and the corresponding sidereal time The right ascension of the mean sun and its determination Given the mean solar time at any instant to find the corresponding sidereal time Given the sidereal time at any instant to find the corresponding mean solar time vii 39. 40 42 44 44 47 48 CONTENTS viii CHAPTER IV INSTRUMENTS AND THEIR USE PAGE. Instruments used by the engineer 31. 50 TIMEPIECES 32. Historical 50 33. Error and rate 51 34. Comparison of timepieces The care of timepieces 52 35. 58 THE ARTIFICIAL HORIZON 36. Description and use 37. 38. Description and theory Uncertainty of the result 39. Historical 40. Influence of imperfections of construction and adjustment Summary of the preceding section 59 THE VEHNIEB 59 60 THE 41 . ENGINEER'S TRANSIT 61 62 ,. 71 42. The 43. Precepts for the use of the striding level Determination of the value of one division of a level 72 The measurement of vertical angles The measurement of horizontal angles The method of repetitions 77 44. 45. 46. 47. 71 level 73 80 81 THE SEXTANT 49. Historical and descriptive The principle of the sextant 50. Conditions 5 88 53. Adjustments of the sextant Determination of the index correction Determination of eccentricity corrections 54. Precepts for the use of the sextant 91 55. The measurement 91 48. 1 . 52. fulfilled 85 86 by the instrument 87 89 90 of altitudes CHAPTER V THE DETERMINATION OF LATITUDE 56. Methods 95 1. 57. 58. 96 Theory Procedure 2. 96 DIFFERENCE OF MERIDIAN ZENITH DISTANCES 59. Theory 60. Procedure . 62. Theory Procedure TALCOTT'S METHOD 97 98 3. 61 MERIDIAN ZENITH DISTANCE ClRCUMMERIDIAN ALTITUDES 99 101 CONTENTS ix ZENITH DISTANCE AT ANY HOUR ANGLE 4. PAGE. 63. Theory 64. Procedure 102 103 5. 104 66. Theory Procedure 67. Influence of an error in time 65. ALTITUDE OF POLABIS . . 105 106 CHAPTEK VI THE DETERMINATION OF AZIMUTH 68. Methods 108 1. 69. Theory 70. Procedure 109 110 2. 71. 72. AZIMUTH OP A CIBCUMPOLAK STAR AT ANY HOUR ANGLE 110 Theory Procedure 112 3. 73. AZIMUTH OF THE SUN AZIMUTH FROM AN OBSERVED ZENITH DISTANCE 74. Theory Procedure 75. Azimuth mark 114 76. Influence of an error in the time 114 113 . of a 113 CHAPTER VII THE DETERMINATION OF TIME 77. Methods 116 1. 78. 79. Theory Procedure 117 llg 2. 80. Theory 81 Procedure . THE ZENITH DISTANCE METHOD THE METHOD OF EQUAL ALTITUDES 118 119 3. THE MERIDIAN METHOD 82. Theory 120 83. Procedure 123 4. THE POLABIS VERTICAL CIRCLE METHOD SIMULTANEOUS DETERMINATION OF TIME AND AZIMUTH 84. Theory 126 85. Procedure 129 ERRATA INDEX J32 PRACTICAL ASTRONOMY FOR ENGINEERS CHAPTER I INTRODUCTION CELESTIAL SPHERE COORDINATES. investigations. The investigations of 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, 1. The results of astronomical the astronomer have shown 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 constitution. 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 coincides 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 In addition to the motion of revolution, the sun, planets, and gravitation. 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 Their diameters vary from about three thousand to nearly million miles. thousand miles. Nevertheless, comparatively speaking, they are small, ninety 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 comparatively 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. PRACTICAL ASTRONOMY The solar and stellar systems are by no means coordinate parts of the On the contrary, the former, vast as it is, is but an insignificant of the latter, for the sun is but a star, not very different on the portion average from the other stars whose total number is to be counted by hun- universe. and the space containing the entire solar system, includand planetoids, is incredibly small as compared with that occupied by the stellar system. To obtain a more definite notion dreds of millions ing sun, planets, ; satellites, 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. of the relative size of the OBJECT : IXTKODUCTION We do know, however, that in many instances 3 two or more stars situated relatively near each other revolve about their common center of gravity thus The discovery and study of these forming binary or multiple systems. 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 obit is only when the utmost refinement of observation 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. jects are so great that is 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 fundamental 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 observers as the distinguishing characteristic of the stars is shown quent use of the expression fixed stars. The nebulae are to be counted by the hundreds of thousand's. by the fre- They conwidely 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 the nebulae through some evolutionary process the details of which are as yet not fully sist of understood. The preceding paragraphs givo the barest outline of the interpretation' which astronomers have been led to place upon the phenomena of the The development of this conception of the structure of the universe forms the major part of the history of astronomy during the last four cert- heavens. PRACTICAL ASTRONOMY Many have turies. its more contributed toward the elaboration of its details, but due to Copernicus, Kepler, and Newton. Although the scheme outlined above is the only theory thus far formulated 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 by 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 significant features are 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 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 few 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 of different brightness, in the northern heavens describe arcs of circles in a counter-clockwise di- 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 rection The surface to which the light-points seem atpoint mentioned above. 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 is 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 will 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 A conthat the position of this object changes with respect to the stars. eastward moves tinuation of the observations will show that it apparently over the surface of the sphere along a great circle at such a rate that an 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 four minutes given star rises does not remain the same, but occurs some entire circuit is INTRODUCTION 5 A star rising two hours after sunset on a h given night will rise approximately l 56 after sunset on the following The average intervals for succeeding nights will be l h 52 m , l h 4&m , l h night. m 44 , etc., respectively. That the stars rise earlier on successive nights shows earlier for each successive night. 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 This amount varies somewhat, being greatest in of rising of the stars. 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 These are rigidly fixes these objects to the surface of the celestial sphere. 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- 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 rection 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 combination 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. Relation of the apparent phenomena to their interpretation. The re3. 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 trace. The existence. celestial The sphere rs purely an optical phenomenon and has no real though differing greatly in distance are all celestial bodies so far from the observer that the eye fails to distinguish any difference in The blue background upon which they seem projected is their distances. due partly to reflection, and party to selective absorption of the light rays by the atmosphere surrounding the earth. As already explained, the stars PRACTICAL ASTKOXOMY 6 are so distant that, barring a few exceptional cases, their individual motions produce no sensible variation in their relative positions, and, even for the On the other hand, exceptions, the changes are almost vanishingly small. the sun, planets, and satellites are relatively near, and their motions produce marked changes mutual distances and in their positions with respect of the sun in the ecliptic is but a reflection of the motion of the earth in its elliptical orbit about the sun. The monthly motion of the moon is a consequence of its revolution about the earth, and the complex motions of the planets are due, partly to their own revolutions about the sun, and partly to the rapidly shifting position of the observer. Finally, the diurnal rotation of the celestial sphere, which at first glance seems to carry with it all the celestial bodies,, is but the result of the axial to the stars. in their The annual motion rotation of the earth. more obvious phenomena of the heavens are concerned no contradiction involved in either of the conceptions which have been devised for the description of their relations. That such is the case arises from the fact that we are dealing with a question concerning changes of relative distance and direction. Given two points, A and B, we can describe the fact that their distance apart, and the direction of the line joining them, are changing, in either of two ways. We may think of A as fixed and B moving, or we may think of B as fixed and A in motion. Both methods are correct, and each is capable of giving an accurate description of the change in relative distance and direction. So, in the case of the celestial bodies, we may describe the variation in their distances and directions, either by assuming the earth to be fixed with the remaining bodies in motion, or by choosing another body, the sun, as the fixed member of the system and describing the phenomena in terms of motions referred to it. The former method of procedure is the starting point for the system of Ptolemy, the Both methods are correct, and hence neither latter, for that of Copernicus. In so far as the there is can give rise to contradiction so Though two ways lie long as the problem remains one of motion. open before us, both leading to the same goal, the by no means a matter of indifference, for one is much more For the discussion of many questions the conception of a fixed earth and rotating heavens affords a simpler method of treatment but, when a detailed description of the motions of the planets and satellites is required, the Copernican system is the more useful by far, although the the geocentric theory presents no formal contradiction unless we pass beyond and relative of case as a motion, attempt consideration of the phenomena If their explanation as the result of the action of forces and accelerations. the of central the earth the makes which body this be done, the conception universe comes into open conflict with the fundamental principles of mechanics. With the heliocentric theory there is no such conflict, and herein choice of route is direct than the other. ; lies the essence of the various so-called proofs of the correctness of the Copernican system. The problems of practical astronomy are among those which can be more simply treated on the basis of the geocentric theory, and we might INTRODUCTION "' have proceeded to an immediate consideration of our subject from this of emphasizing the character primitive stand-point but for the importance of what we are about to do. For the sake of simplicity, we shall make use of ideas which are not universally applicable throughout the science of asbecause it tronomy. W>e shall speak of a fixed earth and rotating heavens so in but, doing, it is imis convenient, and for our present purpose, precise ; outlined above, portant always to bear in mind the more elaborate scheme and be ever ready to shift our view-point from the relatively simple, elemento the more matary conceptions which form a part of our daily experience, must ever be the delight jestic structure whose proportions and dimensions and wonder of the human mind. Relation of the problems of practical astronomy to the phenomena The problems of practical astronomy with which we are of the heavens. concerned are the determination of latitude, azimuth, time, and longitude. The latitude of a point on the earth may be defined roughly as its (a) 4. angular distance from the equator. It can be shown that this is equal to the complement of the inclination of the rotation axis of the celestial sphere to the direction of the plumb line at the point considered. If the inclination of the axis to the plumb line can be determined, the latitude at once becomes known. of a point is the angle included between the vertical rotation axis of the celestial sphere and the vertical the plane containing orientation of the vertical plane through If the the object. plane through the axis of the sphere can be found, the determination of the azimuth of the The azimuth (b) point becomes but a matter of instrumental manipulation. Time measurement is based upon the diurnal rotation of the earth, (c) which appears to us in reflection as the diurnal rotation of the celestial rotation of the celestial sphere can therefore be made the basis measurement. To determine the time at any instant, we have The sphere. time only to find the angle through which the sphere has rotated since some specified initial epoch. As will be seen later, the determination of the difference in longi(d) tude of two points is equivalent to finding the difference of their local times. of The solution of the longitude problem therefore involves the application of the methods used for the derivation of time, together with some means of comparing the local times of the two places. The latter can be accomplished by purely mechanical means, quite independently of any astronomical phenomena, although such phenomena are occasionally used for the purpose. In brief, therefore, the solution of these four fundamental problems can be connected directly with certain fundamental celestial phenomena. Both latitude and azimuth. depend upon the position of the rotation axis of the celestial sphere, the former, upon its inclination to the direction of the plumb line, through it ; the latter, upon the orientation of the vertical plane passing while the determination of time and longitude involve the posi- tion of the sphere as affected by diurnal rotation. PRACTICAL ASTRONOMT 8 A word more, and we are immediately led to the detailed consideration of our subject The solution of our problems requires a knowledge of the position of the axis of the celestial sphere and of the orientation of the : sphere about that axis. We meet at the outset a difficulty in that the sphere and its axis have no objective existence. Since our observations and measurements must be upon things which have visible existence, the stars for example, we are forced to an indirect method of procedure. We must make our measurements upon the various celestial bodies and then, from the known location of these objects on the sphere, derive the position of the sphere and its axis. This raises at once the general question of coordinates and coordinate systems to which we now give our attention. Coordinates and Coordinate Systems. Position is a relative term. cannot specify the position of any object without referring it, either explicitly or implicitly, to some other object whose location is assumed to be known. The designation of the position of a point on the surface of 5. We a sphere is most conveniently accomplished by a reference to two great circles that intersect at right angles. For example, the position of a point on the earth is fixed by referring it to the equator and some meridian The angular distance of the point as that of Greenwich or Washington. from the circles of reference are its coordinates in this case, longitude and latitude. Our first step, therefore, in the establishment of coordinate systems for the celestial sphere, is the definition of the points and circles of reference which will form the foundation for the various systems. The Direction of the Plumb Line, or the Direction due to Gravity, produced indefinitely in both directions, pierces the celestial sphere above in the Zenith, and below, in the Nadir. The plane through the point of observation, perpendicular to the direction of the plumb line, is called the Horizon Plane. Produced indefinitely in all directions, it cuts the celestial sphere in a great circle called the Horizon. Since the radius of the celestial sphere is indefinitely great as compared with the radius of the earth, a plane through the center of the earth perpendicular to the direction of For many purgravity will also cut the celestial sphere in the horizon. poses it is more convenient to consider this plane as the horizon plane. The called the celestial North sphere is pierced by its axis of rotation in two points and the South Celestial Pole, or more briefly, Celestial Pole It is evident from the the North Pole and the South Pole, respectively. relations between the phenomena and their interpretation traced in Section 3 that the axis of the celestial sphere must coincide with the earth's axis of rotation. Great circles through the zenith and nadir are called Vertical Circles. Their planes are perpendicular to the horizon plane. The vertical circle passing through the celestial poles is called the Celestial Meridian, or simply, the Meridian. Its plane coincides with the plane of the terrestrial meridian through the point of observation. vertical circle intersecting the meridian at an angle of ninety degrees The intersections of the meridian and prime called the Prime Vertical. The is vertical with the horizon are the cardinal points, North, West. Small circles parallel to the East, South, and horizon are called Circles of Altitude or Almucanters. Great circles through the poles of the celestial sphere are called Hour Circles. The great circle equatorial to the poles of the celestial sphere is called the Celestial Equator. The plane of the celestial equator coincides with the plane of the terrestrial equator. Small circles parallel to the celestial equator are called Circles of Declination. The already denned as the great circle of the celestial sphere followed by the sun in its annual motion among the stars, is inclined to the The points of intercelestial equator at an angle of about 23 degrees. ecliptic, equator are the Equinoxes, Vernal and Autumnal, respectively. The Vernal Equinox is that point at which the sun in its annual motion passes from the south to the north side of the equator; the Autumnal Equinox, that at which it passes from the north to section of the ecliptic and the celestial the south. The Solstices, midway between the equinoxes are called the Summer and Winter, respectively. The Summer Solstice lies to points on the ecliptic the north of the celestial equator, the Winter Solstice, to the south. The coordinate systems most frequently used in astronomy present certain features in common, and a clear understanding of the underlying principles will greatly aid in acquiring a knowledge of the various systems. At the basis of each system is a Fundamental Great Circle. Great circles perpendicular to this are called Secondary Circles. One of these, called the Principal Secondary, and the fundamental great circle, form the reference circles of the system. The Primary Coordinate is measured along the fundamental great circle from the principal secondary to the secondary passing through the object to which the coordinates refer. The Secondary Coordinate is measured along the secondary passing through the object from the fundamental great circle to the object itself. The fundamental great circle and the principal secondary intersect in two points. The intersection from which the primary coordinate is measured, and the direction of measurement of both coordinates, must be specified. In practical astronomy three systems of coordinates are required. The details are shown by the following table. The symbol used to designate each coordinate is written after its name 'in the table. sometimes more convenient to use as secondary coordinate the distance of the object from one of the poles of the fundamental great circle. Thus in System I we shall frequently use the distance of an object from It is 10 PRACTICAL ASTRONOMY COORDINATE SYSTEMS. SYSTEM COORD1XA TES Point 11 PRACTICAL ASTRONOMT 12 The bending of the light rays by known as Refraction, affects all of amount of the refraction, under which the object is which is the earth's atmosphere, a always small, depends upon the conditions The allowance for its influence is there- observed. made by each individual observer. amount will be discussed in Section 8. fore In the first of observation. phenomenon The the coordinates but azimuth. 1 The method of determining its system, the reference circles are fixed for any given point The azimuth and altitude of terrestrial objects are therefore constant, unless the point of observation is shifted. For celestial bodies, on the contrary, they are continuously varying. The positions of all such objects are rapidly and constantly changing with respect to the circles of reference, as a result of the diurnal rotation. For the nearer bodies, an additional introduced by their motions over the sphere and the changing It appears, therefore, that azimuth and position of the earth in its orbit. altitude are of special service in surveying and in geodetic operations, but complexity is that their range of advantageous application in connection with celestial bodies is limited, for not only are the azimuth and altitude of a celestial object constantly changing, but, for any given instant, their values are different for all points on the earth. But in spite of this disadvantage, altitude, at least, is of great importance. Its determination in the case of a celestial body affords convenient methods of solving two of the fundamental problems with which we are concerned, viz., latitude and time. Since the fundamental circle in the first system depends only upon the direction of the plumb line, the instrument required for the measurment of altitude is extremely simple, both in construction and use. In consequence, altitude is the most The observational part readily determined of all the various coordinates. of the determination of latitude and time is therefore frequently based upon measures of by altitude, the final results being derived from the observed data a process of coordinate transformation to be developed in Chapter II. In the third system, the reference circles share in the diurnal rotation. fixed on the sphere, their motions are so slow that the coordinates of objects, which, like the stars, are sensibly fixed, remain practically constant for considerable intervals of time. Right ascension and Although not absolutely declination are therefore convenient for listing or cataloguing the positions of the stars. Catalogues of this sort are not only serviceable for long periods but can also be used at all points on the earth. The latter circumstance renders right ascension and declination an advantageous means of expressing the positions of bodies not fixed on the sphere. For such objects of time, we have only to replace the single pair of coordinates which suffices for a star, by a series giving the right ascension and declination for equi-distant intervals of time. Such a list of positions is called an Ephemeris. If the time intervals separating the successive epochs for which the coordinates are given be properly chosen, the position can be found for any intermediate 'The azimuth of objects near the horizon is also affected by refraction. change in the coordinate is very small, however. of the The magnitude COORDINATES instant by a process of interpolation. The 13 interval selected for the tabula- determined by the rapidity and regularity with which the coordinates change. In the case of the sun, one day intervals are sufficient, but for the moon the positions must be given for each hour. For the more distant planets, whose motions are relatively slow, the interval can be increased tion is to several days. Collections of ephemerides of the sun, moon, and the planets, together with the right ascensions and declinations of the brighter stars, are published annually by the governments of the more important nations. That issued by our own is prepared in the Nautical Almanac Office at Washington, and bears the title "American Ephemeris and Nautical Almanac." It necessary to examine the character of the variations produced by the slow motion of the reference circles mentioned The mutual attractions of the sun, moon', and the planets produce is in the coordinates above. small changes in the positions of the equator and ecliptic. The motion of the ecliptic is relatively unimportant. That of the equator is best understood by tracing the changes in position of the earth's axis of rotation. As the moves does not remain absolutely parallel to a a conical surface. The change in the but describes given position, 26000 about of the axis takes direction place very slowly, years being required earth in its orbit, the axis initial for it to return to its original position. During this interval the inclination of the equator to the ecliptic never deviates greatly from its mean value of about 23^. Consequently, the celestial pole appears to move over the sphere in a path closely approximating a circle with the pole of the The direction of the motion is counter-clockwise, and ecliptic as center. the radius of the circle equal to the inclination of the equator to the ecliptic. The actual motion of the pole is very complex; but its characteristic features are the progressive circular component which causes it component already mentioned, and a transverse nod back and forth with respect to oscillate or to the pole of the ecliptic. The result is a vibratory motion of the equator about a mean position called the Mean Equator, the mean equator itself slowly revolving about a line perpendicular to the plane of the ecliptic. The motion of the equator combined with that of the ecliptic produces an oscillation of the equinox about a mean position called the Mean Vernal Equinox, which, in turn, has a slow progressive motion toward the west. The resulting changes in the right ascension and declination are divided into two classes, called precession and nutation, respectively. Precession is that part of the change in the coordinates arising from the progressive westward motion of the mean vernal equinox, while Nutation is the result of the oscillatory or periodic mean equinox. The amount motion of the true vernal equinox with respect to the of the precession and nutation depends upon the position For an object on the equator the maximum value of the precession in right ascension for one year is about forty-five seconds of arc or three seconds of time. For stars near the pole it is much larger, amounting in of the star. PRACTICAL ASTRONOMY 14 the case of e The annual precession example, to about <W relatively small, and does not exceed 20" for any of the Polaris, in declination is for . stars. There remains to be considered the effect of the object's and that of the observer. We have already seen own motion how the changes arising from the motion of such objects as the sun, moon, and the planets can be expressed by means of an ephemeris giving the right ascension and declination for equi-distant intervals of time. For the stars the matter is much Their motions over the sphere are so slight as to be entirely inappreciable in the vast majority of cases, and for those in which the change cannot be disregarded, it is possible to assume that the motion is uniform and along the arc of a great circle. The change in one year is called the If the right ascension and declination are given for star's Proper Motion. it is desired to find their values as affected by proper any instant, /, and motion for any other instant f, it is only necessary to add to the given coordinates the products of the proper motion in right ascension and declinasimpler. expressed in years. The position of a star for a given proper motion are therefore all that is required for the determination of its position at any other epoch, in so far as the position is dependent upon the star's own motion. The motion of the observer may affect the position of a celestial object tion into the interval/ initial epoch and /' its two ways First, the actual change in his position due to the diurnal and annual motions of the earth causes a change in the coordinates called Parallactic Displacement. Second, the fact that the observer is in motion at the instant of observation may produce an apparent change in the direction in which the object is seen, in the same way that the direction of the wind, as noted from a moving boat or train, appears different from that when the observer is at rest. The change thus produced .is called Aberration, and is Aberration carefully to be distinguished from the parallactic displacement. not at all his the observer's and upon position, velocity, depends only upon except as position may determine the direction and magnitude of the motion. Parallactic displacement, on the contrary, depends on the distance over which the observer actually moves. For the nearer bodies the parallactic displacement due to the earth's annual motion is large, and is included with the effect of the object's own motion in the ephemeris which expresses its positions. The variation arising from the rotation of the earth on its axis is far smaller, and can always be treated as a correction. In the case of the stars, the distances are so great that the maximum known parallactic displacement due to the earth's annual motion amounts to only three-quarters of a second of arc. For all but a few, a shift in the position of the earth from one side of its orbit to the other, a distance of more than 180,000,000 miles, reveals no measurable change in the coordinates. The displacement due to the earth's rotation is in : of course altogether inappreciable. Parallactic displacement is usually called Parallax, and, of, signifies specifically, when so spoken the correction which must be applied to the observed C O ORDINA TES 1S coordinates of an object in order to reduce them to what they would be were the object seen from a standard position. For the stars, the standard position is the center of the sun for all other bodies, the center of the earth. : Aberration is due to the fact that the velocity of the observer is a quantity For all of appreciable magnitude as compared with the velocity of light. stars not lying in the direction of the earth's orbital motion, the telescope must be inclined slightly in advance of the star's real position in order that rays may pass centrally through both objective and eye-piece of the The star thus appears displaced in the direction of the obinstrument. server's motion. The amount of the displacement is a maximum when the its is at right angles to the direction of the star, and to zero when the two directions coincide. The rotation of the earth equal on its axis produces a similar displacement. The Diurnal Aberration is so direction of the motion minute, however, that observations. The coordinates it requires consideration only in the most refined of the second system possess, to a certain degree, the properties of those of both and altitude, is a coordinate Systems I and III. Hour angle, like azimuth which varies continuously and rapidly, and is dependent on the position of the observer on the earth. The secondary coordinate, declination, is the same as in System III, and the remarks conThe second system cerning it made above, apply with equal force here. of prime importance in the solution of the problems of practical astronomy, it serves as an intermediate step in passing from System I to System III, It is also the basis for the construction of the equatorial or vice versa. is for mounting for telescopes, the form most commonly used in astronomical inves- tigations. 7. Summary. Method of treating the corrections in practice. It is to be remembered, therefore, that the azimuth and altitude of terrestrial objects are constant for a given point of observation, but change as the For celestial objects they observer moves over the surface of the earth. are not only different for each successive instant, but also, for the same instant, they are different for different points of observation. Right ascension and declination are sensibly the same for all points on the earth, and, in consequence, are used in the construction of catalogues and ephemerides. One pair of values serves to fix the position of a star for a long period of time, but for the sun, the moon, and the planets an ephemeris is required. The corrections to which the coordinates are subject are proper motion, precession, nutation, annual aberration, diurnal aberration, parallax, stellar or planetary as the case may be, and refraction. Right ascension and declination are affected by all, but only planetary parallax, refraction, and diurnal aberration arise in practice in connection with azimuth and altitude, and In all cases these three are of these three the last is usually negligible. dependent upon local conditions, and consequently, their calculation and application are them left to the observer. Since it is impracticable to include catalogue and ephemeris positions of right ascension and declination, there remains to be considered, as affecting such positions, proper motion. in PRACTICAL ASTRONOMY 16 precession, nutation, annual aberration, and stellar parallax. The last is so As rarely of significance in practical astronomy that it can be disregarded. it is sometimes necessary to know; their collective effect, and It thus happens sometimes, the influence of the individual variations. that we have different kinds of positions or places, known as mean place, true place, and apparent place. The mean place of an object at any instant is its position referred to for the others, the mean equator and mean equinox of that instant. The mean place is by proper motion and precession. The true place of an object at any instant affected is its position referred to the true equator and true equinox of that instant, that is, to the instantaneous positions of the actual equator and equinox. The true place is equal to the mean place plus the variation due to the nutation. The apparent place of an object at any instant is equal to the true place at that instant plus the effect of annual aberration. It expresses the location of the object as it would appear to an observer situated at the center of the earth. The positions to be found in star catalogues are mean places, and are referred to the mean equator and equinox for the beginning of some year, for example, 1855.0 or 1900.0. Such catalogues usually contain the data necessary for the determination of the precession corrections which must be applied to the coordinates in deriving the mean place for catalogues also contain the value of the proper any other epoch. Modern motion when appreciable. The nutation and annual aberration corrections are found from data given by the various annual ephemerides. The ephemerides themselves contain mean places for several hundred of the brighter stars; but the engineer is rarely concerned with these, or with the catalogue positions mentioned above, for apparent places are also given for the ephemeris stars, and these are all that he needs. The apparent right ascension and declination are given for each star for every ten days throughout the year. Apparent positions are also given by the ephemeris for the sun, the moon, and the planets, Positions for all of these bodies for dates for suitably chosen intervals. intermediate to the ephemeris epochs can be found by interpolation. With this arrangement, the special calculation of the various corrections necessary for the formation of apparent places is avoided entirely in the discussion of all ordinary observations. The observer must understand the origin and significance of all of the changes which occur in the coordinates, in order to use the ephemeris intelligently; but he has occasion to calculate specially only those which depend upon the local conditions affecting the observations, The first we disregard on viz., diurnal aberration, parallax, and refraction. consideration of the engithe account of its minuteness. There remains for neer only refraction and parallax. The following is a brief statement of the methods by which their numerical values can be derived. 8. Refraction. The velocity of light depends upon the density of the medium which it traverses. When a luminous disturbance passes from a medium of one density into that of another, the resulting change in velocity REFRACTION shifts the direction of the wave 17 front, unless the direction of propagation is Stated otherwise, perpendicular to the surface separating the two media. from medium into of a light ray passing one another different density under- goes a change in direction, unless the direction of incidence is normal to The the bounding surface. This change in direction is called Refraction. incident ray, the refracted ray, and the normal to the bounding surface at the point of incidence lie in a plane. When the density of the second medium greater than that of the first, the ray is bent toward the normal. When the conditions of density are reversed, the direction of bending is away from is the normal. The light rays from a celestial object which reach the eye of the observer must penetrate the atmosphere surrounding the earth. They pass from a region of zero density into one whose density gradually increases from the smallest conceivable amount to a maximum which occurs at the surface of rays undergo a change in direction as indicated above. The altitude of all celestial bodies, without sensibly changing their azimuth unless they are very near the horizon. For the case of two media of homogeneous density, the phenomenon of refraction the earth. The effect is to increase the simple; but here, it is extremely complex and its amount difficult of determination. The course of the ray which reaches the observer is affected not only by its initial direction, but also by the refraction which it suffers is at each successive point in its path through the atmosphere. The latter is determined by the density of the different strata, which, in turn, is a function This brings us to the most serious difficulty in the problem, our knowledge of the constitution of the atmosphere, especially in its upper regions, is imperfect. To proceed, an assumption must be made concerning the nature of the relation connecting density and altitude. This, combined with the fundamental principles enunciated above, forms the basis of an elaborate mathematical discussion which results in an expression giving the refraction as a function of the zenith distance of the object, and the temperature of the air and the barometric pressure at the point of observation. This expression is complicated and cumbersome, disadvantages overcome, in a measure, by the reduction of its various parts to tabular form in accordance with a method devised by Bessel. With this arrangement, the determination of the refraction involves the interpolation and combinationof a half dozen logarithms, more or less. Various hypotheses concerning the relation between density and altitude have been made, each of which gives rise to a distinct theory of refraction, of the altitude. for although the differences between the corresponding numerical results are That generally used is due to Gylden. The tables based upon this slight. are known as the Pulkova Refraction Tables, and can be found in the theory more comprehensive works on spherical and When practical astronomy. the highest precision is desired these tables or their equivalent must be used, but for many purposes a simpler procedure will suffice. For example, the approximate expression, PRACTICAL ASTRONOMY 18 983* tan 2', (3) derived empirically from the results given by the theoretical development, 1 can be used for the calculation of the refraction, r, .when the altitude is not than 15. In this expression, b is the barometer reading in inches; /, the temperature in degrees Fahrenheit; z, the observed or apparent zenith distance. The refraction is given in seconds of arc. The error of the result less exceed one second. For rough work the matter can be still further simplified by using mean Fahr. the coefficient values for b and /. For 6=%9.5 inches, and / =50 will rarely of (3) is 57", whence = 57" tan z'. (4) The values of r given by (4) can be derived from columns three and eight of Table I with either the apparent altitude or the apparent zenith distance as argument. For altitudes greater than 20 and normal atmospheric conditions, the error will seldom exceed a tenth of a minute of arc. 9. Parallax. The parallax of an object is equal to the angle at the object subtended by the line joining the center of the earth and the point Fig. 2 Thus, in Fig. 2, the circle represents a section of the earth C is the center of coinciding with the vertical plane through the object. the earth, the point of observation, the zenith, and B the object. The of observation. Z and z are the apparent and geocentric zenith distances, respectively. Their difference, which is equal to the angle /, is the parallax of B. angles z' 'This form was derived by Comstock, Bulletin of the Series, v. i, p. 60. University of Wisconsin, Science PARALLAX We 18 have-the relations z = z' (5) p, (6) altitudes, respectively. The zenith distances and decrease is to increase therefore, parallax, refraction. that the of altitudes, produced by just opposite The parallax depends upon p the radius of the earth, r the distance where ft' and h are the apparent and geocentric effect of of the object the triangle from the earth's center, and the zenith distance z' or 2. From OCB The angle / does not exceed a few seconds of arc for any celestial body therefore excepting the moon. For this its maximum value is about 1. write We /=sin.s'. The horizon, coefficient is p / r, the value of the parallax called the Horizontal Parallax. /=/ The value of p Denoting (7) when its the body value sin*'. varies with the distance of the object. is the by/ we have (8) It is tabulated American Ephemeris for the sun (p. 285), the moon (page IV of For the sun, however, the each month), and the planets (pp. 21&-249). in so that we use its is mean value of 8"8, whence p slight may change in the /= 878 sin z'. (9) error of this expression never exceeds 0''3. The values of / corresponding to (9) can be interpolated from columns four and nine of Table I. The For approximate work the solar parallax is conveniently combined with The difference of the two corrections refraction given by (4). can be derived from the fifth and tenth columns of Table I with the apparent the mean altitude or the apparent zenith distance as argument. The preceding discussion assumes that the earth is a sphere. On this the parallax in azimuth is zero. Actually, the earth is spheroidal in form, whence it results that the radius, />, and consequently the angle OBC, do not, in general, coincide with the vertical plane through B\, for the basis line does not point toward the center of the earth, except at the The actual parallax in zenith distance and at points on the equator. poles is therefore slightly different from that given by (9). and in addition, there plumb PRACTICAL ASTRONOMT 20 a minute component affecting the azimuth. The influence of the spheroidal form of the earth is so slight, however, that it requires consideration only in the most precise investigations. Finally, it should be remarked that the apparent zenith distance used for the calculation of the parallax is the observed zenith distance freed from is that is, of the two corrections, refraction is to be applied first-. zenith distance thus corrected serves for the calculation of the parallax. refraction The ; For the first system of coordinates, therefore, and the limits of precision here considered, the influence of both refraction and parallax is confined to the coordinate altitude, or its alternative, zenith distance. Hour angle, right ascension, and declination are all affected by both refraction and parallax, but, as these coordinates do not appear as observed quantities in the problems with which we are concerned, the development of the expressions which give the corresponding corrections TABLE I. omitted. is MEAN REFRACTION AND SOLAR PARALLAX Barometer, 29.5 h' in.; Thermometer, 50 Fahr. CHAPTER II FORMULAE OF SPHERICAL TRIGONOMETRY TRANSFORMATION OF COORDINATES GENERAL DISCUSSION OF PROBLEMS. 10. The fundamental Transform- formulae of spherical trigonometry. ations of coordinates are of fundamental importance for the solution of most The relations between of the problems of spherical and practical astronomy. the different systems should therefore receive careful attention. The more complicated transformations require the solution of a spherical triangle, and, because of this fact, a brief exposition of the fundamental formulae of spherical trigonometry is introduced at this point. Let ABC, Fig. 3, be any spherical triangle. Denote its angles by A, B, and C'; and its sides by a, b, and c With the center of the sphere, 0, as origin . , XY construct a set of rectangular coordinate axes, XYZ, such that the plane Let the reccontains the side c, and the axis passes through the vertex B. tangular coordinates of the vertex Cbe x, y, and z. Their values in terms of X the parts of the triangle and the radius of the sphere are x rcos<z, = r sin a cos B, z =. r sin a sin B. y Construct a second set of axes, coinciding with the side c, and the (10) with the origin 0, the XY plane X axis passing through the vertex A. Let XYZ 1 at , the coordinates of preferred to this system be x\ /, and /. y = = d =. x The second ing the first r cos We then have i>, r sin b cos A, r sin b sin A, set of rectangular axes can be derived from the about the Zaxis through the angle 21 c. first The coordinates by rotat- of the first PRACTICAL ASTRONOMY 22 system can therefore be expressed in terms of those of the second by means of the relations x = x'cos c y sin = x sin c +y cos z = z. y c, (12) c, Substituting into equations (12) the values of x,y, 2, x', y, and z from (to) common factor r, we obtain the desired relations and (n), and dropping the sin sin cos a a cos B a sin B cos b cos c = cos b sin c sin b sin -\- sin b sin c cos sin b cos c cos A, A, (13) (14) A. (15) These equations express relations between five of the six parts of the spherical triangle ABC, and are independent of the rectangular coordinate axes introduced for their derivation. Although the parts of the triangle in Fig. 3 are all less than 90, the method of development and the results are These relations are the fundageneral, and apply to all spherical triangles. mental formulae of spherical trigonometry. From them all other spherical trigonometry formulae can be derived. They determine without ambiguity a side and an adjacent angle of a spherical triangle in terms of the two remaining sides and the angle included between them, provided the algebraic sign of the sine of the required side, or of the sine or cosine of the required angle, be known. Otherwise there will be two solutions. Equations (i3)-(is) are conveniently arranged as they stand if addition- For use with subtraction logarithms are to be employed for their calculation. the ordinary logarithmic tables, they should be transformed so as to reduce the addition and subtraction terms in the right terms (Num. Cal. pp. 13 and 14). members of (13) and (14) to single Aside from the case covered by equations (i3)-(i5), two others occur in connection with the problems of practical astronomy, viz., that in which the given parts are two sides of a spherical triangle, and an angle opposite one of them, to find the third side; and that in which the three sides are given, to find one or more of the angles. The first of these can be solved for those cases which arise in astronomical practice by a simple transformation of (13), the details of which will be considered in connection with the determination of latitude. A solution for the third case can also be found by a rearrange- ment of the terms of (13). cos Thus, A a cos b cos c = cos r-T sin b sin c Similar expressions for the angles . B and Ccan . (16) be derived by a simple permu- tation of the letters in (16). Equation (16) affords a theoretically accurate solution of the problem; but, practically, the application of expressions of this form is limited on account of the necessity of determining the angles from SPHERICAL TRIGONOMETRY For numerical calculation it is important to have formulae such that the angles A, B, and C can be interpolated from their tangents (Num. The desired relations can be derived by a transformation Col. pp. 3 and 14). 12 and 16-18), giving of (Chauvenet, Spherical Trigonometry, their cosines. (16), .,' = sin 5 sin = (s-a) B + + C and can be derived Similar expressions for b s c). (a three angles of a spherical When the of letters of the a (17). by permutation it is advantageous to introduce triangle are to be determined simultaneously, in which # the auxiliary K, defined by the relation _ sin (s-a) sin (s-6) sin (s-c) , , g Sin J Substituting (18) into (17), we find tanj^ A = sin K -. (19) r. -. (s- a) y2 B and tan J4<7are similar in form. the Collecting results, complete formulae for the calculation of the three a of angles spherical triangle from the three sides are The expressions tan s Form s - = % (a + b + c a, s - sin (s-a) b, ). and s-c, and check by sin (s-l>) sin (s-c) . . sin s If Check: tan Two one ]^ If %A tan y, B tan The ambiguity solutions are possible. of the half-angles of the triangle is known. y2 C = is IS -. removed if the quadrant of Relative positions of the reference circles of the three coordinate systems. The transformation of the coordinates of one system into those of another requires a knowledge of the relative positions of the reference circles 11. of the various systems. In the case of Systems I and II the principal secondary circles coincide are inclined to each other at an fundamental circles by definition. The great which and to the is constant complement of the latitude of the angle equal of can be derived from Fig. 4, of The this statement observation. place proof which represents a section through the earth and the celestial sphere in the PRACTICAL ASTRONOMY 24 plane of the meridian of the point of observation, O. The outer circle represents the celestial meridian, and the inner, the terrestrial meridian of 0, the latter being greatly exaggerated with respect to the former. and TV are the Z P and P', the poles of the celestial sphere;/ and/', the zenith and the nadir; HH' and EE\ the lines of intersection of the planes of poles of the earth; horizon and equator, respectively, with the meridian plane. The plane of the celestial equator coincides with that of the terrestrial equator, which cuts the terrestrial meridian in ee. Fig. 4- Now, by definition the arc eO measures the latitude, ip, of the point But, (21) whence H'E = 90 (22) which was to be proved. It thus appears that the second system can be derived from the first by rotating the first about an axis passing through the east and west points, through an angle equal to the co-latitude of the place. It is to be noted, further, that Arc ZP 90 iff = Co-latitude of,* 0, (23) and Arc HP = <f. (24) From (21) and (24) it follows that the latitude of any point on the earth equal to the declination of the zenith of that point. It is also equal to the altitude of the pole as seen from the given point. Systems II and III have the same fundamental great circle, viz., the celestial equator. The principal secondary of the third system does not mainis RELATIVE POSITION OF COORDINATE SYSTEMS tain a fixed position with respect to that of the first, but rotates clockwise direction as seen from the north side of the equator. uniformly 25 in a Let Fig. 5 represent an orthogonal projection of the celestial sphere upon is the north celestial pole; the plane of the equator as seen from the North. the celestial equator; and V, of intersects meridian where the the M, point therefore measures the instantaneous the vernal equinox. The arc position of the principal secondary of the third system with respect to that of P MBV This arc is equal to the hour angle of the vernal equinox, or the 0. It is called the Sidereal Time ascension of the observer's meridian. right thus have the following important definition: The sidereal time at any instant is equal to the hour angle of the the first. = We vernal equinox at that instant. It is also equal to the right ascension of the observer's meridian at the instant considered. It follows, therefore, that the third system can be derived from the second by rotating the second system about the axis of the celestial sphere through an angle equal to the sidereal time. Finally, the third system can be derived from the first by rotating the first into the position of the second, and thence into the position of the third. Briefly stated, the transformation of coordinates involves the determination of the changes arising in the coordinates as a result of a rotation of the various systems in the manner specified above. It is at once evident that the transformation of azimuth and altitude into hour angle and declination requires a knowledge of the latitude; of hour angle and declination into right ascension and declination, a knowledge of the sidereal time; while, to pass from azimuth and altitude to right ascension and declination, both latitude and sidereal time It is scarcely necessary to add that the reverse transformations are required. demand the same knowledge. Transformation of azimuth and zenith distance into hour angle 12. and declination. The transformation requires the solution of the spherical The essential part of Fig. I is reproduced in Fig. 6 triangle ZPO, Fig. I, p. n. upon an enlarged scale. parts of the triangle ZPO An inspection of the notation of p. can be designated as shown in Fig. n 6. shows that the PRACTICAL ASTRONOMY '26 Assuming the latitude, y>, to be known, it is seen that the transformation involves the determination of the side it and the adjaquestion 90 cent angle t in terms of the other two sides, 90 and z h, and 90 y> the angle 180 A included between them. Equations (i3)-(i5) are directly = in applicable, and it is = only necessary to make the following assignment of parts: = 90 = 3, * c 90 <p. a A = i8oA, *, (25) Fig. 6. The substitution of (25) into (13), (14), and (15) gives 3 sin To adapt cos 8 cos t cos 3 sin t = cos z sin sin z cos = cos z cos + sin z sin = sin z sin A. <p ip <f> cos A, (26) <p cos A, (27) (28) these formulas for use with the ordinary logarithmic tables, the and M, defined by auxiliary quantities m m sin M= mcosM = are introduced (Num. sin z cos cos A, z, Cat. p. 14). Substituting these relations into (26) and (27) and collecting results, have for the calculation m sin M = m cos M = cos 3 sin t cos 3 cos t sin 3 In formulae (26)-(28) unknown quantities, t we have and 3; sin z cos cos we A, z, = sin z sin A, = m cos - M), = m sin (tp-M). (29) (<p three equations for the determination of two equations are given for the determin- in (29), five ORDER OF SOLUTION 27 In both cases one more condition ation of the four unknowns, M, m, /, and 3. is available than is required for the theoretical solution of the problem, a point of great practical importance, as the numerical solution. The order first it affords a means of testing the accuracy of M and m from the is as follows: First, determine Whatever the values of z and A, there will always be two and m satisfying these equations. For one, m will be pos- of solution two equations. M pairs of values of It is immaterial, so far as the final values of / itive; for the other, negative. and 8 are concerned, which of the two solutions we adopt. For simplicity, is always positive. This makes the algebraic signs however, we assume that of sin M and cos M the m same as those of the right-hand and second equations, respectively, of members of the first The difference of the logarithms of the right-hand members of these equations equals log tan M, from which the angle determined, the quadrant being fixed by a consideration of the algebraic signs of any two of the three functions, sin M, cosJ/, and tanM. (29). Mh M are interpolated with a single opening of the M, and log cos The difference of the last two must equal log tan M, which affords a The subtraction of log sin from log m sin partial check. gives log m. The addition of this result to log cos must agree with the value of log mcosM from the second of (29), which gives a second partial check. The values M, log sin table. M of M and m M M thus derived are to be substituted, along with z and A, into the members of the last three of (29) for the completion of the calculation. The left members of the third and fourth of (29) are of the same form as the first two, which makes it possible to determine /and cos 8 by an application of the process employed for finding m and M, care being taken to apply the checks at the points indicated above. The algebraic sign of cos 3 is necright-hand essarily positive, since 3 must always lie between +90 and 90, which fixes the quadrant of /. It is to be noted that this limitation upon the sign of cos<J removes the ambiguity existing in the solution of the general spherical triangle which was mentioned on p. 22. The hour angle, /, and log cos 3 having been found, the next step is the determination of log sin 3 from the last of (29). The values of log cos 3 and log sin 3 must correspond to the same angle. This affords a third partial check. The determination of 3 and the application of the check can be accomplished in either of two ways: We may interpolate 3 from the smaller of the two functions logcosiJ and logsincJ, and check by comparing the other function with the value interpolated from the tables with the calculated d as argument; or we may interpolate d from log tan 3, which is found by subtracting log cos 3 from log sin 8. With the value of d thus derived, log sin 3 and log cos <J are interpolated from the table. The interpolated values must agree with those resulting from the last three equations of (29). The former method is shorter; the latter, more precise in the long run, although not necessarily so in any specific case. In practice, the first method is usually sufficient. In applying the checks it is to be noted that the accumulated error of calculation (Num. Cal. pp. 4 and 12) may produce a disagreement of one, and in rare instances, of two units in the last place of decimals. Great care must bf PRACTICAL ASTRONOMY exercised with the algebraic signs of the trigonometric functions and in assigning the quadrants of the angles. Otherwise, an erroneous computation may check. The check apparently quantities must agree both in absolute magni- tude and algebraic sign. The calculation of t and 8 from equations (26)-(28) with the aid of additionsubtraction logarithms is accomplished by an application of the method used for the solution of the last three of (29). The only differences which occur are to be found in the details of the combination of the quantities which enter into the right members Example 1. of the two groups of equations. For a place of observation whose latitude is 38 56' 51", the azimuth of an its zenith distance 62 37' 49". Find the corresponding hour angle object is 97 14' 12" and and declination. The calculation for equations (z6)-(28), using addition-subtraction logarithms, appears column; that for equations (29), made with the ordinary tables, is in the second in the first column. For the first, <5 is derived from log sin 3, which, in this case, is smaller than log cos d. In the calculation of (29), 3 is determined from log tan 3. The arguments tor the check quantities, sin 5 and cos 5, need not ordinarily be written down. They are inserted here In order to illustrate the application of the control. The abbreviation /o^-is not prefixed to the arguments, although the majority of the numbers appearing Its omission saves time and produces no confusion. in the computation are logarithms. TRA NSFORMA T1ONS 29 Transformation of hour angle and declination into azimuth and 13. zenith distance. The transformation can be effected by solving (29) in the reverse order to that followed in Section 12. It is better, however, to use equations of the same form as those appearing in the preceding section, thus reducing the two problems to the same type. As before, two sides and the included angle are given, to find the remaining side. With the following assign- ment of parts = = 90 c = 90 a we find by substituting into sin A B z, b cos 2 z cos A , I= and as before, A, (30) d sin (15), <p -\- cos o cos sin <Jcos^> + cosd sin cososin/ sin cos/, (31) ycost, (32) <f> 1 (33) . These are of the same general form as same principle 180 <p, (13), (14), = = t, = (26), (27), and (28). Applying the we derive N = sin N = cos d cos A = cos d sin sin z cos A = sin N), cos z = n cos N). n sin 3, n cos sin z sin /, /, (34) (<p (<p The two groups (3i)-(33) and (34) give the required transformation. The former can be used with addition-subtraction logarithms; the latter, with the A comparison of these equations with groups (26)-(28) and shows that the same arrangement of calculation can be used for both transformations. The unknowns are involved in the same manner in both cases, with the exception that the sine and cosine of z are interchanged in the ordinary tables. (29) left members of (3i)-(34) as compared with the corresponding functions of d in (26)-(2 9 ). In the solution of (3i)-(33) and (34), the quadrant of A is fixed by the fact that sin z is necessarily positive, since z is always included between o and + 180. This eliminates the ambiguity attached to the solution of the general spherical triangle. Transformation of hour angle into right ascension, and vice 14. versa. Since the coordinate declination is common to Systems II and III, the transformation of the coordinates of one of these systems into those of the other requires only a knowledge of the relation between hour angle and right ascension. In Fig. 5, p. 25, let be the intersection of the hour circle through celestial body with the celestial equator. then have by definition B We any PRACTICAL ASTRONOMY 30 MB = Hour angle of object, = = Right ascension of object, = 9= Sidereal time, Arc Arc VM Arc MVB t whence (35) a. (36) Equations (35) and (36) express the required transformations. The same can be derived from Fig. I, p. II, the point /, in this figure, correspond- result ing to B in Fig. 5. Example 2. In a place of observation whose latitude is 38 58' 53", the hour angle of an h m 8 31' 47". Find the corresponding azimuth and object is 2o i9 4i8, and its declination zenith distance. The calculation by equations (3l)-(33) i* in the first column; the second. t = 20" I9 m 41-8 = -8 = 38 ? 3 tint = 304 31' 47" 58 53 55' 27" that by equations (34), in TRANSFORMATIONS Example when What 3. the sidereal time By equation the right ascension of an object Is Example (35) What 4. the sidereal time By equation whose hour angle it I7 h 2i m 34!6, 2i h i4 ni 52!8? is / when 31 is = 2i b i4 n 52!8 = 17 21 34.6 = 3 S3 18.2, Ans. the hour angle of an object whose right ascension is 8 h i2 m 34!8, is (36) a = & 6m 28'7 = 8 12 34.8 53 53-9> Ans. t Transformation of azimuth and altitude into right ascension and 15. declination, or vice versa. These transformations are effected by a combination of the results of Sections 12-14. For the direct transformation, determine / and 8 by (26)-(28) or (29), and then a by (35). For the reverse calculate / by (36), and then A and z by (3i)-(33) or (34). Example 5. What is the right ascension of the object whose coordinates, at the sidereal lime I7 h 2i ln i6*4, are those given in Example I? The hour angle found in the solution of Example i by equations (26)-(28) is 4 h47 m 46'4. h m This, combined with ff I7 2i i6!4 in accordance with equation (35^, gives for the required ll nl right ascension I2 33 3o!o. = Example 6. At a place whose latitude is 38 38' 53", what are the azimuth and zenith distances of an object whose right ascension and declination are 9 k 27 ra i4'2 and 8 31 '47", reh m . spectively, the sidereal time being 5 46 56o? = = 2oh i9 m 4i8. We have, further, t 8 (36;, / These quantities are the same as those appearing in Example By equation tions (39) gave A = 300 10' 29", z = 69 31 '47" 2. and The = <f 38 38' 53". solution by equa- 42' 30". Given the latitude of the place, and the declination and zenith 16. distance of an object, to find its hour angle, azimuth, and parallactic angle. We have given three sides of the spherical triangle ZPO, Fig. 6, p. 26, to find the three angles, the parallactic angle being the angle at the object. The is parallactic angle frequently required is not used in practical in engineering astronomy, although its value astronomy proper. Equations (20) are directly applicable for the solution of the problem. Assigning the parts of the triangle as in (30), and, further, writing the angle C q parallactic angle, we have for the calculation. = = PRACTICAL ASTRONOMY 32 = a = 90 c = 90 = (a + b + b z, s Check: = 2 3, 'A + (s-a) sin + (s-i>) -c) (s K Object { ^ In engineering all that is tan T, /A tan J4 ' cot ^ l 2 tan /^ ^ % A, % q /, s, Yi sin (J-^)' of meridian, } = w sin (j-<r) (j-a)sin (j-) sin (.y-<z)' Check: if, c), in = q= f ^" d } { ge K sin (j-c)' quadrant astronomy the determination of the hour angle, t, is usually For this case it is simpler to use equation (17). The required. formulae are = a b 2, s Check: tan . = 90 =y 2 (5 % d, (a + = in = 90 c + + c). - b) + (*-*) in b - a) f - (s - (s sin s sin (s <f, c) = s, (38) (j-,) - a) y2 t is to be taken in the first or second quadrant according as the obwest or east of the meridian at the time of observation. For those cases in which the object is more than two and one-half or three hours from the meridian, equation (16) written in the form where ject is cos* cos - z sin will usually give satisfactory results. trol upon the value of (39) is / given by readily calculated by 17. <?sin cos 3 cos (38). means a> , (39) if In any case, (39) affords a valuable conThe numerator of the right member of of addition-subtraction logarithms. Application of transformation formulae to the determination of It was latitude, azimuth, and time. the fundamental problems of practical shown in Section 4 that the solution of astronomy requires the determination of the position of the axis of the celestial sphere and the orientation of the sphere as affected by the diurnal rotation. In practice this is accomplished indirectly by observing the positions of various celestial bodies with respect to the horizon, the observed data being combined with the known position of the bodies on the sphere for the determination of the position of the sphere itself. The means for effecting the coordinate transformation hereby implied are to be found in the formulas of Sections 12-16. Although the most advantageous determination of latitude, azimuth, and time requires a modification of these formulas, it is, nevertheless, easy to see that the solution of the various problems is within our grasp, and that the TRANSFORMA TIONS GENERAL DISCUSSION Example 7. For a place whose latitude Is 3S56'si", find the hour angle, azimuth, and parallactic angle of an object east of the meridian whose declination and zenith distance are 8 16' 14" and 54 16' 12", respectively. Equations (37) are used for the solution, which is given below in the column on the left. only the hour angle were required, equations (38) or (39) would be used. As an illustration of the application of these formula;, the problem is also solved on this assumption. The first ten lines of the computation for (38), being the same as that for The (37), are omitted. If remainder of the calculation for (38) occupies the upper part of the right-hand column. The solution by (39) is in the lower part of this column. The object is rather too near the meridian for the satisfactory use of equation (39), although it happens that the resulting value of the hour angle agrees well with that from (37) and (38). 8 38 54 98 b 6' 16 51 '2 16 14 c 5i 3 9 23 35 35 s 101 47 48 - a 47 3" 36 s-6 3 3i 34 - c 50 44 39 i sin (s - a) sin (i-*) 9.86782 sin (s 9.88892 -c) cosec s 0.00927 2 log /T A" log 8-55491 tan 9.27746* %t cot Yi A tan q cot A tan % t 8.78890 % K cosec s sin 8.78890 In 9.88892 (s-6) (i-c) cosec (s-a) 14" 2i s tan )4 1 5 cosec tan* y s t 0.13218 0.00927 8.81927 9.40964, Ck. 165 35' 46" 33i ii 32 PRACTICAL ASTRONOMT 34 intimate relation to all of the other kinds of time, so that, if the sidereal time has been found, the determination of the others becomes but a matter of calculation. would be complicated. It is simpler to deterfor the calculation of each that one or and A, assuming separately, <p, both of the others are known. - a. Let it For example, equation (31) is a function of 2, <J, <p, and / Practically, such a solution mine = be assumed that the zenith distance of a star of known right ascension and declination has been measured and that the time of observation has been noted. The substitution of the resulting data into (31) leads to the determination of the only remaining unknown, namely, the latitude, (p. Again, the elimination of z from (32) and (33) gives an expression for =d- A be assumed that <p and 6 are known. <p, The azimuth of a star of known right ascension and declination can therefore be calculated. The calculated azimuth applied to the observed difference in .azimuth of star and mark gives the azimuth of the mark. as a function for d, and t Let a. it and (39) express the hour angle, t, as a function of the zenith distance of a star of known right ascension and dec- Finally, equations (38) z, (f, and 3. lination be culated. If measured Equation in a place of (35), in known the form t hour angle can be calthen gives the sidereal time of latitude, the = + a, observation. The solutions thus outlined require, for the determination of latitude, a for the determination of time, a knowledge of the latFor the first two, time and itude; and, for azimuth, both time and latitude. If each is latitude, it might appear that the methods proposed are fallacious. knowledge of the time; required for the determination of the other, how can either ever be determined? The explanation is to be found in the fact that the formulas can be arranged in such a way that an approximate value for either of these quantities suffices for the determination of a relatively precise value of the other. Thus, a mere value of the latitude, as to the time will lead to a accurate guess relatively which, in turn, can be used for the determination of a more precise value of the time. The process can be repeated as many times as may be necessary to secure the desired degree of precision. The principle involved in the procedure In thus outlined is called the Method of Successive Approximations. numerical investigations it is of great importance. The method amounts,, practically, to replacing a single complex process by a series, consisting of Ordinarily, the success of the repetitions of some relatively simple operation. method depends upon the number must be made in of repetitions or approximations which If the convergence is order to arrive at the desired result. rapid, so that one or two approximations suffice, the saving in time and labor as compared with the direct solution is frequently very great. Indeed, in some instances, the method of successive approximations is the only method of procedure, the direct solution being impossible as a result of the complexity of the relation connecting the various quantities involved. GENERAL DISCUSSION The general method of procedure latitude, time, 35 for the solution of the problems of There remains the formulation a detailed development, we must and azimuth has been outlined. But, before proceeding to in its theoretical aspects the different kinds of time, their definition and their relations. Chapter III will be devoted to this must also consider the various astronomical instruments that question. of the details. consider the subject of time We find application in engineering astronomy their characteristics and the con- ditions under which they are employed, since the nature of the data obtained through their use will influence the arrangement of the solutions. Chapter IV is therefore devoted to a discussion of various astronomical instruments. In arranging the details of the methods for the determination of latitude, time, and azimuth, it is to be remembered that the various problems are not merely to be solved, but they are to be solved with a definite degree of This requirement precision, and with a minimum expenditure of labor. renders the question one of some complexity, for the precision required may vary within wide limits. For many purposes approximate results will suffice, and it is then desirable to sacrifice accuracy and thus reduce the labor involved. On the other hand, in astronomical work of the highest precision, no means should be overlooked which can in any way contribute toward an elimination or reduction of the errors of observation and calculation. The problems with which we have to deal therefore present themselves under the most diverse conditions, and, if an intelligent arrangement of the methods is to be accomplished, one must constantly bear in mind the results which will be established in the two following chapters, as well as those already obtained in the discussion of the principles of numerical calculation. CHAPTER III TIME AND TIME TRANSFORMATION The 18. basis for the basis of time measurement. The rotation of the earth is the measurement of time. Since motion is relative, we must specify the object to which the rotation obvious that is referred. By referring to different objects, it is several different kinds of time. Actually, the rotation referred to three different things: the apparent, or true, sun, a we may have of the earth is mean sun, and the vernal equinox. In practice, howturn the matter about and take the apparent diurnal rotations of these objects with reference to the meridian of the observer, considered to 'be fixed, as the basis of time measurement. have, accordingly, three kinds of time fictitious object called the ever, we We Apparent, or True, Solar Time, Mean : Solar Time, and Sidereal Time. Apparent, or True, Solar Time=A.S.T. 19. time at any instant is The apparent, or true, solar equal to the hour angle of the apparent, or true, sun at that instant. The between two successive transits of the apparent, or true, sun is called an Apparent, or True, Solar Day=A. S. D. The instant of transit of the apparent sun is called Apparent Noon=A. N. interval across the same meridian It is The In astronomical practice the apparent solar day begins at apparent noon. subdivided into 24 hours, which are counted continuously from o to 24. earth revolves about the sun in an elliptical orbit, the sun itself occupying one of the foci of the ellipse. The earth's motion is such that the radius vector with the sun sweeps over equal areas in equal times. Since the connecting distance of the earth from the sun varies, it follows that the angular velocity of the earth in its orbit is variable. Hence, the angular motion of the sun it along the ecliptic, variable. The which is but, a reflection of the earth's orbital motion, motion along the ecliptic projection into the equator of the is also is like- wise variable, not only because the ecliptical motion is variable, but also on account of the fact that the angle of projection changes, being o degrees at the solstices, and about 23^ degrees at the equinoxes. Apparent solar time is are nor a therefore, not, apparent solar days of uniformly varying quantity, the same length. The adoption everyday life of such a time system for the regulation of the affairs of it many inconveniences, the first of which would would bring with be the impossibility of constructing a timepiece capable of following accurately the irregular variations of apparent solar time. On this account there has been devised a uniformly varying time, based upon the motion of a fictitious body called the 20. mean Mean sun. Solar Time=M. S. T. The mean sun is an imaginary body supposed to move with a constant angular velocity eastward along the equator, such that it completes a circuit of the sphere in the same time as the apparent, or true, sun. Further, the mean sun is so chosen that its right ascension differs as little as possible, on the average, from that of the true sun. 36 DEFINITIONS 37 The Mean Solar Time at any instant is equal mean sun at that instant. The interval between two successive transits of same meridian The is called a Mean Solar mean sun instant of transit of the Mean solar time of the same length. Day=M. is S. called to the hour angle of the the mean sun across the D. Mean Noon=M. N. a uniformly varying quantity and all mean solar days are Mean solar time is the time indicated by watches and clocks, is generally, throughout the civilized world, and the unit for the measurement of time. mean In astronomical practice the solar mean solar day day begins at is the standard mean noon. It is subdivided into 24 hours which are numbered continuously from o to 24. The astronomical date therefore changes at noon. But since a change of date during the daylight hours would be inconvenient and confusing for the affairs of everyday life, the Calendar Date, or Civil Date, is supposed to change 12 hours before the transit of the mean sun, i.e. midnight preceding the astronomical at the change of date. Further, in most countries, the hours of the civil mean solar day are not numbered continuously from o to 24, but from o to 12, and then again from o to 12, the letters A. M. or P. M. being affixed to the time in order to avoid ambiguity. For example the civil date 1907, Oct. 8, ioh A. M., is h to the The astronomical day astronomical date, 1907, Oct. 7, 22 equivalent . Oct. 8 did not begin until the calendar. mean sun was on the meridian on Oct. 8 of the From the manner of definition, it is evident that at solar time for different places not on the same meridian place were to attempt mean is would To avoid this difficulty the time of the same meridian. all If each own local different. to regulate its affairs in accordance with solar time, confusion traffic. any instant the mean its especially in connection with railway points within certain limits of longitude use arise, The meridians selected for this purpose are an exact multiple of 15 degrees from the meridian of Greenwich, with the result that all timepieces referred to them indicate at any instant the same number of minutes and seconds, and differ among themselves, and from the local mean solar time of the meridian of Greenwich, by an exact number of hours. The all system thus defined is called Although, theoretically, Standard Time. all a standard meridian use the local points within lY? degrees of longitude of mean solar time of that meridian, actually, the separating adjacent regions whose standard times differ by one boundaries hour are quite irregular. The standard meridians for the United States are 75, 90, 105, and 120 degrees west of Greenwich. The corresponding standard times are Eastern, Central, Mountain, and Pacific. These are slow as compared with Greenwich mean solar time by 5, 6, 7, and 8 hours, respectively. 21. Sidereal Time. The sidereal time at any instant is equal to the hour angle of the true vernal equinox at that instant. (See p. 25.) The interval between two successive transits of the true vernal across the same meridian is called a Sidereal Day=S. D. equinox PRACTICAL ASTRONOMY 38 The instant of Noon=S. transit of the true vernal equinox is called Sidereal N. Since the precessional and nutational motions of the true equinox are not uniform, sidereal time is not, strictly speaking, a uniformly varying quantity, but practically it may be considered as such, for the variations in the motion of the equinox take place so slowly that, for the purposes of observational astron- omy, all sidereal days are of the same length. of sidereal time in the transformation of the coordinates The importance of the second system into those of the third, and vice versa, has already been shown in Sections 11 and 14. It also plays an important role in the determination of time generally, for sidereal time is more easily determined than either apparent or The mean solar time. usual order of procedure in time determination is as follows: Every observatory possesses at least one sidereal timepiece whose error is determined by observations on stars. The true sidereal time thus obtained is transformed into mean solar time by calculation, and used for the correction of the mean solar timepieces of the observatory. Certain observatories, in particular the United States Naval Observatory at Washington, and the Lick Observatory at Mt. Hamilton in California, send out daily over the wires of the various tele- graph companies, series of time signals which indicate accurately the instant of mean noon. These signals reach every part of the country, and serve for the regulation of watches and clocks generally. 22. The Tropical Year. Several different kinds of years are employed astronomy. The most important are the tropical and the Julian. The Tropical Year is the interval between two successive passages of the mean sun through the mean vernal equinox. Its length is 365.2422 M. S. D. During this interval the mean sun makes one circuit of the celestial sphere from in equinox to equinox again, in a direction opposite to that of the rotation of the follows that during a tropical year the equinox must complete 366.2422 revolutions with respect to the observer's meridian. therefore have the important relation: sphere itself, whence it We One Tropical Year=365-2422 M. S. D.=366.2422 S. D. (40) In accordance with a suggestion due to Bessel, the tropical year begins when the mean right ascension of the mean sun plus the constant h part of the annual aberration is equal to 280 or i8 40"'. The symbol for this instant is formed by affixing a decimal point and a zero to the corresponding at the instant year number; thus for 1909, the beginning of the tropical year is indicated by This epoch is independent of the position of the observer on the earth 1909.0. and does not, in general, coincide with the beginning of the calendar year, although the difference between the two never exceeds a fractional part of a day. 23. The For chronological purposes the use of a year involving day would be inconvenient. That actually used has its a decree promulgated by Julius Caesar in 45 B. C. which ordered that Calendar. fractional parts of a origin in THE CALENDAR 39 the calendar year should consist of 365 days for three years in succession, these The extra day of the fourth year was to be followed by a fourth of 366 days. sixth the twice introduced by counting day before the calends of March in the such years were long distinguished by the Years. The years of designation bissextile, although they are now called Leap the this With Years. are Common average length of arrangement 365 days Roman system. In consequence the calendar year was 365^ days. This period is called a Julian Year, and the calendar based upon it, the Julian Calendar. The difference between the Julian and the tropical years is about ii m . In order to avoid the gradual displacement of the calendar dates with respect to the seasons resulting from the accumulation of this difference, a slight modification in the method of counting leap years difference was introduced in 1582 by Pope amounts approximately to three Gregory XIII. The accumulated days in 400 years, and, as the Julian year is longer than the tropical, the Julian calendar falls behind the seasons by this amount. Gregory therefore ordered that the century years, all of which are leap years under the Julian rule, should not be counted as such unless the year numbers are exactly divisible by 400. At the same time it was ordered that 10 days should be dropped from the calendar in order to bring the date of the passage of the sun through the vernal equinox back to the 2ist of March, where it was at the time of the Council of Nice in 325 A. D. The Julian system thus modified is called the Gregorian Calendar. All years revised rule for the determination of leap years is as follows: the whose numbers are exactly divisible by four are leap years, excepting century These are leap years only when exactly diiisible by four hundred. All years. other years are common years. The average length of the Gregorian calendar The In the year differs from that of the tropical year by only 0.0003 day or 26". modern system the extra day in leap years appears as the 2Qth of February. The Gregorian calendar was soon adopted by all Roman Catholic countries and by England in 1752. Russia and Greece and other countries under the dominion of the Eastern or Greek Church, still use the Julian Calendar, which, at present, differs from the Gregorian by 13 days. 24. time at Given the local time at any point, to find the corresponding local From the definitions of apparent solar, mean solar, point. and sidereal time, it follows that at any instant the difference between two local times is equal to the angular distance between the celestial meridians to which But this is equal to the angular distance between the the times are referred. any other geographical meridians of the two places, Let Tt TV L We i.e. their difference of longitude. = the time of the eastern place, = the time of the western place, = longitude difference of the two places, then have the relations: Te = Tw +L (41) PRACTICAL ASTRONOMY 40 Equations (41) are true whether the times be apparent mean solar, solar, or sidereal. Example Given, Columbia 8. Greenwich mean solar time I2 h I4 m 16:41, find the corresponding mean solar time. 7V= i2 h H m 16141 L Te 9 18.33 23 34.74 = 6 = 18 Example 9. Given, Greenwich mean sponding Washington mean solar time. Te = 1907, Ans. solar time 1907, Oct. 6 3 h I4 m 21", find the corre- Oct. 6 = TV = 1907, Oct. L 3 h I4 m 21" 5 8 16 5 22 6 5 Ans. h h s Given, central standard time 1907, Oct. 12 6 i8 o A.M., find the corresponding Greenwich mean solar time, astronomical and civil. Example 10. TV = L Te 1907, Oct. 12 Oct. II = = 1907. Oct. 12 Oct. 12 6 h 18 o" A.M. 18 18 o astronomical o o 18 o astronomical \ 18 o P.M. / 600 civil n Example 11. Given, central standard time 1907, Oct. corresponding Columbia mean solar lime, civil and astronomical. Te L = 1907, Oct. = 7\v= II oh 3 m i6'i8 9 18.33 oh 3 i6i8 P.M. find the P.M. 1907, Oct. ii ii 53 57.85 A.M. Oct. 10 23 53 57.85 astronomical civil ^^ j / 25. Given the apparent solar time at any place, to find the corred a, sponding mean solar time, and vice versa. From equation (36), t and the definitions of mean solar and apparent solar time, we find = =d = M. S. T. A. S. T. A. S. T. = E = M. R. A. of M. R. A. of A. S., S. whence M. The S. T. R. A. of M. R. A. of A. S. S. difference S. T. - A. S. T. (42) Equation of Time. The equation of time varies irregularly m It is somethe throughout year, its maximum absolute value being about i6 times positive, and sometimes negative, since the right ascension of the apparent is called the . sometimes smaller and sometimes greater than that of the mean sun. The ascension of the apparent sun is calculated from the known orbital motion right of the earth. The right ascension of the mean sun is known from its manner sun is TIME TRANSFORMATION This data of definition. suffices for the calculation of 41 , whose values are tabu- In the American Ephemeris they are given for instants of Greenwich apparent noon on page I for each month, and for Greenwich mean noon, on page II. The former page is used when apparent time is converted into the corresponding mean solar time, and the lated in the various astronomical ephemerides. latter when apparent solar time E is found from a given mean solar time. to be not given in the American Ephemeris, but the column is headed by a precept which indicates whether it is to be added to or subtracted from the given time. Values of E for times other than The algebraic sign of containing its values is Greenwich apparent noon and Greenwich mean noon must be obtained by interThis operation is facilitated by the use of the hourly change in E polation. in the columns headed "Difference for i Hour," which immediately folprinted low those containing the equation of time. If the time to be converted refers to a meridian other than that of Greenwich, the corresponding Greenwich time must be calculated before the interpolation is made. Note that for each date the difference of the right ascension of the apparent, or true, sun in column two of page II, and the right ascension of the mean sun in the last column of the same page, is equal to the corresponding value of E, in accordance with the definition. Example 12. Given, Greenwich apparent solar time 1907, Oct. 15 corresponding Greenwich mean solar time. for Gr. A. N. 1907, Oct. 15 in Change E Example 13. E during (to be subtracted 2h 6m 12' from A. Gr. A. S. T. 1907, Oct. 15 Gr. M.S. T. 1907,001.15 mean Given, Greenwich corresponding Greenwich S. T.) Change M. N. in ' -f- 56*75 1.20 26 13 57.95 12.06 i 52 14.11 i M. h p. 164) Ans. i h 52"" 14511, find the -)- S. T.) 1907, Oct. 15 i Gr. A. S. T. 1907, Oct. 15 2 1.07 13 57.95 52 6 14.11 12.06 standard time 1907, Oct. 20 Example Given, corresponding Columbia apparent solar time. central 14. C. S. T. 1907, Oct. 20 nh L Columbia M. S. T. Gr. M. S. T. E for Gr. in M. N. 9 18.3 1907, Oct. 19 23 8 53.9 5 18 12.2 1907, Oct. 20 58.63 14 be added to Columbia M. S. T.) 15 S. T. 1907, Oct. 19 23 23 Columbia A. n i8 m I2!2 1907, Oct, 20 E during 5 h i8 m 12* (Eph. p. 165) 56588 13 52"" 14" M.S.T. E (to (Eph. solar time 1907, Oct. 15 1907, Oct. 15 E during E (to be added to Change 6 m 12506, find the apparant solar time. E for Gr. Gr. m I3 2h + Ana. h i8 m 1252 A.M., find the A.M. astronomical (Eph. 2.39 i.o 54.9 Ans. p. 165) PRACTICAL ASTRONOMT 42 Example Given Columbia apparent solar time 1907, Oct.. 19 23 h 23 m 15. f4"9, find the corresponding central standard time. Columbia A. T. 1907, Oct. 19 23 Gr. A. S. T. 1907, Oct. 20 for Gr. A. N. 1907, Oct. 20 5 S. h E Change in JE h during 5 33 m S. T. 5459 33 13.2 14 58.52 15 i.o 1907, Oct. 19 23 8 53.9 9 i8.3 1907, Oct. 20 II 18 12.2 L C. S. T. 18.3 +2.50 13' E (to be sub. from Columbia A. S. T.) Columbia M. 23 69 L A.M. Ans. Relation between the values of a time interval expressed in and sidereal units. Equation (40) is the fundamental relation If we let the units of mean solar and sidereal time. connecting f 26. mean solar = the value of any interval / in mean 7m = the value of /in sidereal units, A we find from solar units, (40) =/m + = - = /s /m Ia (43) 365^422 (44) 366.2422 Writing ' IT 11T= _ 366.2422 365.2422 (43) and (44) become 7S 7m Assuming 7m = 24 b we find 24" o Similarly, m by supposing 7S h 24 o = /m + Ill/m =7 from 0:000 = 24 h (45) 1 17s 5 (46) (45) = 24" m M. S. we obtain from (46) m O5ooo Sid. = 23 3 565555 Sid. h M. 56"" 45091 S. Hence Gain of on M. Gain of 6 on M. S. T. in i S. T. in i M. S. S. = IIl24 = 2365555 D. = 1124 = 235.909 D. h RELATION OF SIDEREAL AND SOLAR INTERVALS 43 and further on M. Gain of S. T. in I M. hour hour S. Gainof0onM.S.T.ini S. For many purposes these expressions imate relations: be replaced by the following approx- may 1124 = = (l 1/60), IIIi =10. (i 1/70), =10 (i l/6o), IIl24 h III 4 m 4 (i 1/70), = IIIi" = 918565 = III =9.8296 = 0*016 = Error 0.081 Error = 0.0006 Error = 0.0037 Error value Equations (45) and (46) may be used for the conversion of the value its units into in solar mean of a time interval expressed corresponding The calculations are most conveniently made in sidereal units, and vice versa. II by Tables II and III printed at the" end of the American Ephemeris. Table the of III those Table III/m, contains the numerical values of II/s, while gives arguments being the values of the factors of first ment which H/ and /s and 7m, respectively. It will be observed that III/m indicate the table, and the second the argu- to be used for the interpolation. In case tables are not available the conversion can be based is upon equations (47) or (48), or more simply, upon (49), provided the highest precision is not required. Example 16. Given the mean solar interval toMS" 21*20, find the equivalent sidereal 1 interval. By Eq. /m (45) I6"1 18 21*20 2 40.72 21 1.92 III/m = fs The = = 16 calculation of III/m by the third of (49) /m = 16*306 is (Eph. Table Ans. III) as follows: = 163106 = 2.33 Ill/m = 160.73 = 2 m 40!73 IO"/m I/7oXio*/m The value thus found differs onlyo'oi from that derived from Table III of the Ephemeris. Example 17. Given the sidereal interval 20" 28 m 42*17, find the equivalent Interval. By Eq. /, (46) II/S /m The = 2O 11 = = 2O calculation of II/S by the last of (49) /, = 20*478 io*/s 3 25 is 21.29 2O.88 (Eph. Table Ans. as follows: = 204*78 1/60X10*78= II/s 28 m 42*17 = 3.41 201.37 = 3 m 2 "37 II) mean solar PRACTICAL ASTRONOMT 44 27. Relation between mean solar time and the corresponding sidereal time. In Section 14 it was shown that the relation connecting the hour angle of an object with the sidereal time t= is 6 a where a represents the right ascension of the object. to the mean sun, we Applying this equation find M=6 R (50) M R in which its hour represents the right ascension of the mean sun, and angle. The latter, however, is equal by definition to the mean solar time. Equation (50) therefore expresses a relation between mean solar time and the cor- responding sidereal time, which can be made the basis for the conversion of the one into the other. The transformation requires a knowledge of R, the right ascension of the mean sun, at the instant to which the given time refers. now turn our attention to a consideration of the methods which are avail- We able for the determination of this quantity. The right ascension of the mean sun and its determination. It works on theoretical astronomy that the right ascension of the mean sun at any instant of Greenwich mean time is given by the expression 28. is shown in RG = 18" 38 m 45 ! 836 + (236:555 X 365.25)' + osooooo93* + nutation in right ascension, a in It (51) reckoned in Julian years from the epoch 1900, Jan. oa o h Gr. M. T. thus appears that the increase in the right ascension of the mean sun is not which t is proportional to the increase in the time. This, in connection with equation sidereal time is not a uniformly varying quantity, a fact already indicated in Section 21. The nutation in right ascension oscillates between limits strictly (50), shows that s which are approximately-}- I s and I with a period of about 19 years. Its change in one day is therefore very small, and, as the same is true of the term involving 2 t in (51), it follows that the increase in the right ascension of the mean sun one mean solar day is sensibly 236. B 555- From equation (50) it is seen that the gain of sidereal on mean solar time during any interval is equal to the increase in during that interval; and, indeed, we have exact numerical agreement between the change in the latter for one mean solar day, as given by equation (51), and the gain of the former during the same period as shown by the in R of (47). From this it follows that the methods given in Section 26, including Tables II and III of the Ephemeris and the approximate relations (49), can equally well be applied to the determination of the increase in R, provided only that the interval for which the change is to be calculated is small enough to first render the variations in the last two terms of (51) negligible. RIGHT ASCENSION OF MEAN SUN 45 To facilitate the solution of problems in which R is required, its precise numerical values are tabulated in the various astronomical ephemerides for every day in the year. In the American Ephemeris they are given for the instant of Greenwich mean noon, and are to be found in the last column of page II for each month. If these tabular values be represented by Ro, and if Ri. represent the right ascension of the mean sun at the instant of mean noon for a point whose longitude west of Greenwich is L, it follows from the preceding paragraph that RL L = R + IIIZ, O (52) equal to the time interval separating mean noon of the place from the preceding Greenwich mean noon. Further, the value of ./? at any mean time, M, at a point whose longitude west of Greenwich is L is given by for is R= XL + HIM, (53) + (54) or R=R + IIIZ HIM. Equations (52) and (53), or their equivalent, (54), R suffice for the determi- any instant at any place when the value of Ro for the preceding mean noon is known. For a given place the term IIIZ is a constant. Its value can be calculated once for all, and can then be added mentally to the value of Ro as the latter is taken from the Ephemcris. The quantity HIM may be nation of at M derived from Table III of the Ephemeris with as argument. If an Ephemeris is not available the values of R can still be found; approximately at least, by the use of Tables II-IV, page 46. The first of these contains Ro computed from (51) for the date Jan. o for each of the years these Roo and by neglecting the variations in the last two 1907-1920. Denoting terms of (51) we have for Greenwich mean noon of any other date the values of R = R00 + where D indicates the preceding Jan. o. number of mean The month question. IV. value of to the The D = Ro + may be tabular UIL (55) solar days that Substituting (55) into R UW + have elapsed since the (54). 111(1) + M). (56) obtained from Table III by adding the day of the the name of the month in number standing opposite M is conveniently expressed in decimals of a day by means of Table value thus found is to be combined with D. If the precise value of be used, the uncertainty in R derived from (56) will be only from the neglect of the variation in the last terms of (51). If care be taken to count D from the nearest Jan. o the error will never exceed o.'3 or 8 III, viz., 236. 555, that arising PRACTICAL ASTRONOMY 46 TABLE II RIGHT ASCENSION OF THE MEAN SON FOR THE EPOCH JAN. o d o h GR. M. T. Year TRANSFORMATION OF MEAN SOLAR INTO SIDEREAL TIME 47 D> This requires that for 183* the negative value of Table III be employed, for Roo l\\z following Jan. o. value of with the together 0-4. somewhat greater uncertainty is permissible, the result may be more m 1/70) for III. If D be reckoned from the expeditiously found by using 4 (i o as nearest Jan. above, the corresponding error will not exceed 3". If a Example 8 h 2i m 14-00 18. Find the right ascension of the mean sun for the epoch 1907, June 16 Columbia M. S. T. By Equation /. = 6h Af=8 9 m 21 i8'33 ll\L 14.00 lllAf (D + M) = i67<)348 m 4 (D + M) = 6697392 m 1/70 X4 (> + M) = 9TS63 Example 2 6m 2<ji 19. m 25*'0 i 0.67 = = R=5 (54) 34 i 36 22.34 48. By Equation (Tables III (Eph. p. 93) (Eph. Table III) (Eph. Table III) Ans. 1 1 (56) and IV; ^? o (1907) lllL III(Z> -f- = i8 h 36 = M) = R= 10 5 Columbia M. = = = (Table 0-5 i 0.7 59 49.7 3 5' Find the right ascension of the mean sun for the epoch = The = Sh ^?o II) 1909, Sept. 21 S. T. -ioi <1 +o li = = + M) = 8io(Table8lIIandIV)^?oo (1910) i8 h 37 5-1 III/.. i 0.7 35 2.1 loodtgo 4oo-?76o IIl(Z Jf= S-725 precise value given by (54) is I2 h m 3 6 12 34 Ans. find the = M+R R II) 5'2i. 29. Given the mean solar time at any instant to sponding sidereal time. From equation (50) we find Introducing the value of (Table corre- (57) from (53) we have = M+i + UIM, (58) where RL = Ro + lllL. Equations (59) and (58) solve the problem. Equation (58) may be interpreted as follows: of the mean sun at the preceding mean noon (59) ^L is the right ascension for a place in longitude L west of therefore also equal to the hour angle of the vernal equinox i.e. at that instant, to the sidereal time of the preceding mean noon at the is the mean time interval since preceding mean noon, place considered. Now Greenwich. It is M is the equivalent sidereal interval. The right member of (58) therefore expresses the sum of the sidereal time of the preceding mean noon and the number of sidereal hours, minutes, and seconds that have elapsed and by (45) M+IIIAf PRACTICAL ASTRONOMT 48 In other words since noon. is it the sidereal time corresponding to the time, M, by the equation. In case the Ephemeris is not at hand, mean as indicated R may be obtained from (56) and substituted into (57) for the determination of 0. The uncertainty in the sidereal time thus found will be the same as that of derived from (56). Oftentimes a rough approximation for 6 is all that is required. In such R cases the following, designed for use at the meridian of Columbia, 0= The first term in i8"37'?7 + M + 4m member the right i//o) (i (D formula of this is + M). useful : (60) the average value of is m plus the constant term IIIL, which for Columbia may be taken equal to i The expression can be adapted for use at any other meridian by introducing the derived from appropriate value of IIIL. The maximum error in the value of J?oo . is (60) i. m 7. Example 20. Given Columbia mean solar time i6 h 27 32517 on 1909, Nov. 16, find the corresponding sidereal time. (58) and (59) i6 h 27"" 32517 By equations M= Xo III/, = IS = 39 0.67 2 42.23 10 55.05 111M= = 8 By equations D-\-M = m 4 (Z> + A/) = 1/70 X 4 m OD + M)=- 45 d +o M= = IIIL = lll(D + M) = = Koo 44*314 i77-?256 2.532 M= m Aas. (56) and (57) d 686 By equation 4 (i 39-98 i 1/70) (Z> + .W) = # = i6 h 27"" 32:2 18 37 5.1 i 2 8 54 10 0.7 43.4 55 Ans. (60) i8 h 2 37T7 27.5 54-7 8 10. 5 16 Ans. Given the sidereal time at any instant to find the corresponding solar time. We make use of equation (50), viz. 30. mean M=6 Substituting as in Section 29 R we have M=O RL IUM or X I. (61) TRANSFORMATION OF SIDEREAL INTO MEAN SOLAR TIME 49 member by member, and dropping Multiplying equations (45) and (46), factor 7m ft we find the common + (I Combining this we with (61) III)(I II) = I find M= d RL 11(0 RL), (62) where, as before, =R + RL III/. (63) Equations (63) and (62) solve the problem. Equation (62) is susceptible of an interpretation similar to that given (58) in the preceding section. is the given sidereal time, and RL the sidereal Since time of the preceding mean noon, 6 RL is the sidereal interval that has elapsed since noon. To find the equivalent mean time interval we must, in accordance with equation (46), subtract from d ^L the quantity 11(0 RL). The right member of (62) therefore expresses the number of mean solar hours, minutes, and seconds that have elapsed since the preceding mean, noon, solar time corresponding to the given Example 21. Given, 1908, sponding central standard time. May 12, Columbia By equations 9 = RL = RL = RL) = jii (62) sidereal time h 7 m the mean 19*27, find the corre- and (63) 3 20 25.46 46 53.82 M= 3 34-o 21 43 19.72 = 9 18.33 9 52 38.05 L= i jm 19*28 21 C. S. T. i.e. 0. (Eph. Table A.M. May II) 13. Ans. CHAPTER IV INSTRUMENTS AND THEIR USE Instruments used by the engineer. The instruments employed by the for the determination of latitude, time and azimuth are the watch or engineer The the artificial horizon, and the engineer's transit or the sextant. chronometer, 31. following pages give a brief account of the theory of these instruments and a statement of the methods to be followed in using them. The use of both the engineer's transit and the sextant presupposes an under- standing of the vernier. tachment is is In consequence, the construction and theory of this atand sextant treated separately before the discussion of the transit undertaken. TIMEPIECES 32. Historical. Contrivances for the measurement of time have been used since the beginning of civilization, but it was not until the end of the sixteenth century that they reached the degree of perfection which made them of service in astronomical observations. The pendulum seems first to have been used as a means of governing the motion of a clock by Biirgi of the observatory of Landgrave William IV at Cassel about 1580, though it is not certain that the principle employed was that involved in the modern method of regulation. How- may be, the method now used was certainly suggested by Galileo about but Galileo was then near the end of his life, blind and enfeebled, and it 1637; was not until some years later that his idea found material realization in a clock ever this constructed by his son Vincenzio. It remained for Huygens, however, the Dutch physicist and astronomer, to rediscover the principle, and in 1657 give it an application that attracted general attention. Some sixty years later Harrison and Graham devised methods of pendulum compensation for changes of temperature, which, with important modifications in the escapement mechanism introduced by in 1713, made the clock an instrument of precision. Since then its devel- Graham in design and construction has kept pace with that of other forms of astronomical apparatus. The pendulum clock must be mounted in a fixed position. It can not be transported from place to place, and it does not, therefore, fulfill all the requirements that may be demanded of a timepiece. By the beginning of the opment eighteenth century the need of accurate portable timepieces had become pressing, not so much for the work of the astronomer as for that of the navigator. The most difficult thing in finding the position of a ship is the determination of longi- At that time no method was known capable of giving this with anything more than the roughest approximation, although the question had been attacked The by the most capable minds of the two centuries immediately preceding. matter was of such importance that the governments of Spain, France, and the tude. Netherlands established large money prizes for a successful solution, and in 1714 that of Great Britain offered a reward of 20,000 for a method which would give the longitude of a ship within half a degree. 50 With an accurate portable timepiece, TIMBI'lECES 51 which could be set to indicate the time of some standard meridian before beginning a voyage, the solution would have been simple. Notwithstanding the stimuIn 1735 Harrison lus of reward no solution was forthcoming for many years. succeeded in constructing a chronometer which was compensated for changes of temperature; and about 1760 one of his instruments was sent on a trial voyage to Jamaica. Upon return its variation values of the longitudes based on its was found to be such as to bring the readings within the permissible limit of error. The ideal timepiece, so far as uniformity is concerned, would be a. body moving The in practice this can not be realized. under the action of no forces, but modern timepiece of but falls precision is a close approximation to something equivalent, short of the ideal. Thus far it has been impossible completely to nullify the effect of certain influences which affect the uniformity of motion. Changes in temperature, variations in barometric pressure, and the gradual thickening of the oil lubricating the mechanism produce irregularities, even when the skill of No timepiece is perfect. is exercised to its utmost. can say only that some are better than others. Further, it is impossible to set a timepiece with such exactness that it does not differ from the true time the designer and clockmaker We by a quantity greater than the uncertainty with which the latter can be determined. Thus it happens that a timepiece seldom if ever indicates the true time; and, in general, no attempt is made to remove the error. The timepiece is started under conditions as favorable as possible, and set to indicate approximately the true time. It is then left to run as it will, the astronomer, in the meantime, directing his attention to a precise determination of the amount and the rate of change of the error. These being known, the true time at any instant is easily found. 33. Error and rate. The error, or correction, of a timepiece is the quantity which added algebraically to the indicated time gives the true lime. The error of a timepiece which is slow is therefore positive. If the timepiece is fast the algebraic sign of its correction is negative. The error of a mean solar timepiece is denoted by the symbol J7"; of a To designate the timepiece to which the correction sidereal timepiece, by Jt). refers subscripts may be added. Thus the error of a Fauth sidereal clock may Sometimes be indicated by Jti f of a Negus mean time chronometer, by J7*N it is convenient to use the number of the timepiece as subscript. If 6' be the indicated sidereal time at a given instant, and JO the cor. ; responding error of the timepiece, the true time of the instant = 0' + JO'. The analogous formula for a mean solar timepiece The be (64) is T= T + JT'. daily rate, or simply the rate, of a timepiece during one day. will (65) is the change in the error . PRACTICAL ASTRONOMY 52 If the error of a timepiece increases algebraically, the rate is positive; if it negative. The symbols fid and dT with appropriate suband mean solar timescripts are used for the designation of the rates of sidereal The hourly rate,>. the change during one hour, is somepieces, respectively. decreases, the rate is times more conveniently employed than the daily rate. It is convenient, but in no wise important, that the rate of a timepiece should be small. On the other hand, it is of the utmost consequence that the rate should be constant; for the to which reliability of the instrument depends wholly upon the degree this condition is fulfilled. Generally it is impossible to determine by observation the error at the instant must therefore be able to calculate its for which the true time is required. If the rate value for the instant in question from values previously observed. We is constant this can be done with precision; otherwise, the result will be affected by an uncertainty which will be the greater, the longer is the interval separating the epochs of the observed and the calculated errors. If Jt} and Jt)' be values of the observed error for the epochs / and /', the daily rate will be given by which /' /must be expressed in days and fractions of a day. The rate having thus been found, the error for any other epoch, t". may be calculated by the formula in JQ" Example The 22. error of a sidereal + sidereal time, and 5 33510 Feb. 14 at 7^6 sidereal time. We have J0 on = + 5 m 27-61, /' Equation (66) then gives $0 To 1909, Feb. J0'= / = t" i i d n, 5^2 3 t' clock was -f +5 27561 on 1909, Feb. 3, at 6'.'4 and the correction on 6^4 and =7 0:69, d 22l'8 which = is 7< 1 95. the required value of the rate. we have 7*16, = i4 d (67) t') at 5^2; find the daily rate, + 5 m 33"> = + 5H9/7-9S = find the error for Feb. 14, whence by equation = J6' + 3d(t" d 7^6 nd 5^2 = 3'! 2 l'4 = 3<?i, (67) 40" '-= + 5 m 3351 + 3.1 X 0569 = + 5"' 35524. Ans. 34. Comparison of timepieces. It is frequently necessary to know the time indicated by one timepiece corresponding to that shown by another. The determination of such a pair of corresponding readings involves a comparison of the two To make such a comparison the observer must be able accurately timepieces. to follow, or count, the seconds of a timepiece without looking at the instrument. It is desirable, moreover, that he should be able to do this while engaged with other matters, such as entering a record in the observing book, etc. CLOCK COMPAR/SONS 53 Pendulum clocks usually beat, or tick, every second and chronometers, every The beats of the ordinary watch are separated by an interval of a With each beat the second hand of the timepiece moves fifth of a second. forward by an amount corresponding to the interval separating the beats a whole second space for the clock, a half second space for the chronometer, and ; half second. second for the watch. two timepieces coincide, observer has only to pick up the beat from at the other and note the hour, minute, and time on the first. After noting the reading a fifth of a If the beats of a comparison is easily made. The one, then, following mentally, look second corresponding to a definite of the second, the observer should look again at the first before dropping the count, to make sure that the indicated number of seconds and the count were in agreement at the instant of comparison If the beats of the timepieces do not coincide, and it is desired to obtain a com- parison with an uncertainty less than the beat interval, the observer must estimate from the sound the magnitude of the interval separating the ticks. He will then note the hour, minute, second, and tenth of a second on the second timepiece corresponding to the beginning of a second on the first. When a watch is to be should be taken from the compared svith a clock or a chronometer, the count The tenths of a second on a watch corresponding latter. to the beginning of a second on the clock or chronometer may be estimated by noting the position of the watch second hand with respect to the two adjacent second marks at the instant the beat of the clock or chronometer occurs. The comparison will then give the hour, minute, second, and zero tenths on the clock or chronometer corresponding to a certain hour, minute, second, and tenth of a. second on the watch. and a mean solar timepiece are to be compared, a very precise be obtained by the method of coincident beats. It was shown in Section 26 that the gain of sidereal on mean solar time is about ten seconds per hour, If a sidereal result may The ticks of a solar and a sidereal timepiece, each beating seconds, must therefore coincide once every six minutes. If one of the timepieces beats half seconds, the coincidences will occur at intervals of three minutes. comparison is made by noting the times indicated by the two instruments or one second in six minutes. A If carefully made, the uncertainty of the comparison will not exceed one or two hundredths of a second. at the instant the beats coincide. On 1907, Oct. 29, five comparisons of a watch were made with the Fauth Laws Observatory. The means of the comparisons are Or = iS h 23 m o'oo, and 7 w = 4 h 3 nl i6i2 P.M. The error of the Fauth clock was 29172, and the longitude west of Greenwich is 6 h 9 m Find the error of the watch referred to central standard Example 23. sidereal clock of the 1 18*33. time. From Of and J(t r find by (64). The sidereal time is then to be transformed into C. S. T. by (62) and the first of (41;. The resulting C. S. T. compared with 7"w gives the error of the watch. 54 PRACTICAL ASTRONOMY CLOCK COMPARISONS The second and 55 third comparisons are reduced by the method used for Ex. 23. The into C. S. T. are omitted. The two values of J7"c.n present details of the conversion of a satisfactory agreement. Given thirty comparisons of a Wallham watch and a Bond sidereal intervals of one minute; to find the rate per minute of the watch referred to the chronometer, a precise value of the watch time corresponding to the first chronometer reading, and the average uncertainty of a single comparison. Example 26. chronometer made at The interval between any two chronometer readings minus the difference between the corresponding watch readings is the loss of the watch as compared with the chronometer during the interval. The quotient of the loss by the interval in minutes is a value of the relative rate per minute. Thus, if = interval between two chronometer times, = interval between two watch times, R = relative rate of watch per minute, /c 7w then The solution of the first part of the problem may therefore be accomplished by grouping the comparisons in pairs and applying equation (a). The mean of the resulting values of will then be the final result. The selection of the comparisons for the formation of the pairs K requires careful attention if the maximum of precision is to be secured. To obtain a criterion for the most advantageous arrangement, consider the resultant error of observation in when derived from equation (a). Denoting the influence of the errors in the observed watch R times upon the interval 7W by e we find for the error of /? Since c is independent of the length of the interval separating the comparisons, from (b) that the precision of R increases with the length of this interval. it follows R It is desirable for the sake of symmetry in the reduction that the separate values of should be of the same degree of precision; and it is important to arrange the calculation so that any irregularity in the relative rate will be revealed. The reduction will then give not only the quantitative value of the final result, but at the same time will throw light upon the reliability of the instruments employed. We are thus led to the following grouping of the comparisons: i and 16, 2 and 17, 3 and and 30; or, in general, the ath comparison is paired with the(is The )th. fourth column of the table gives the values of 7W corresponding to this choice. The first of these is derived by subtracting the first from the sixteenth; the second, by subtracting the + 18 ....... 15 T from the seventeenth, and soon. The 15 values of 7 substituted into equation (a), together with the constant value 7C 15, would give 15 separate values for R. The first of these would depend upon data secured during the first 15 minutes of the observing period; second 7"w = the last, upon those obtained during the last 15 minutes; while the intermediate values of /,' would correspond to various intermediate 15-minute intervals. Any irregularity in the rate will therefore reveal itself in the form of a progressive change in the separate values of /?. But, since 7C is assumed to be constant throughout, equation (a) shows that constancy of 7W will be quite as satisfactory a test of the reliability of the timepieces as constancy in R. It is not necessary, therefore, to calculate the separate values of the relative rate; and for the der- we adopt the simpler procedure of forming the mean of the values we then substitute into (a) with /c = I5 m We thus find mean 7W = I4 m 57*65, whence the mean relative rate of the watch referred to the chronometer is 05157 per minute of ivation of the final result of 7W which , chronometer time. . PRACTICAL ASTRONOMT 56 WATCH AND CHRONOMETER COMPARISON No. CLOCK COMPARISONS 57 would then have given a precise value of the watch time corresponding to the first chronomThe given problem may be reduced to this case by correcting each watch eter reading. reading by the effect of the rate during the interval separating it from the first observation. To accomplish .... in this we have only 29^?; or, In general, to the column five to add to the readings, th reading, in order, the quantities o/?, i)/f. ( The \R, iff, values of these corrections are of the table, and the watch times, corrected for rate, in column six. These two places of decimals in order to keep the errors ot calculation small as results are given to compared with the errors of the values of TV, io h 25 m 25579, The mean of observation. is the m required precise watch reading corresponding to the first chronometer reading, o' 45 o'oo. To obtain a notion of the uncertainty of a single comparison, consider the corrected watch has been used in applying the corrections for rate, and readings, TV- If the true value of if the true value of the first watch reading were known, the actual error of this and of each of 1 R the remaining readings could at once be found by forming the difference between the true value and each of the corrected watch times. The average of the errors would then indicate ingly, we R and But the true values of the precision of the comparisons. not known and cannot be found. of the first comparison are We must therefore proceed as best we may; and, accorduse for the true relative rate the value calculated above, and for the true value of watch reading, the mean of all the corrected readings. The differences between each corrected watch time and the mean of them all are called residuals. The residuals will differ but little from the corresponding errors, for the calculated value of and the mean TV will the first R differ but little from the quantities they are taken to represent. Although the average of the residuals will not exactly equal the average of the errors, it may be accepted, nevertheless, as a measure of the precision of the observations; for, barring a constant systematic error, it is evident that the more accurate the observations, i.e. the smaller their variations among them- selves, the less will be the average residual. Denoting the residuals by r, and the mean of the corrected watch times by A/ . The A of values of i< formed valuable control may in = A/ accordance with 7"w (0 column of the table. shown that if the exact value algebraic sum must be zero. (Num. It is of the residuals, their M however, an approximation for uals will equal the negative value of the remainder the value used as a mean. p. 17.) we have (c) are in the last be applied at this point. Af be used for the formation Comp. , If, is in easily used, the algebraic sum of the residthe division which gives as quotient In the present case the algebraic sum of the residuals is -fo.oi; the remainder is o.or, which checks the formation of the mean and ihe residuals. The average residual, without This we may accept as the average uncertainty of a single regard to algebraic sign, is 0509. comparison. The principles illustrated in the preceding reduction find frequent application in the treatment of the data of observation. The example is typical and the methods followed in the discussion should receive careful attention. In particular, the grouping of the observations for the determination of the mean value of R should be examined; and the student should investigate for himself the precision of the result when such combinations of the comparisons as 15 I and 2, and 16; 2 and 3, etc, are Example 27. .... 29 and 30; employed i and 2, 3 and 4, .... 79 and 30; i and 30, 2 and 28, .... in place of that actually used. To determine the average uncertainty of a single comparison of two time- pieces by the method of coincident beats. Ten successive coincidences of the beats of a Bond sidereal chronometer with those of a time clock are taken as the basis of the investigation. The method used for the reduction is similar to that employed in Ex. 26. The comparisons are in the second and third columns of the table. Since the chronometer beats halt-seconds and the clock seconds, the interval between the successive coincidences is that required for the clock to lose Gregg & Rupp mean PRACTICAL ASTRONOMY 58 Denote the true value of this interval by I. To ex0:5 as compared with the chronometer. hibit the influence of" the errors of observation we find what the clock readings would have This is done by subtracting all been made at the same instant as the first. The numerical values of the corrections are from the readings, in order, o7, if, 2/, g7. The value to be in column five, and the reduced clock readings themselves, in column six. used for 7 is one-fifth of the average of the intervals between the th and the (-j-5)th clock readings. The individual values of these intervals are in column four. Their mean is n m I4' 55!2, whence 7= 2 59!O4. The variations in the values of 7" represent the influence of the errors of observation. The average residual for the reduced clock readings is 2*94, which been had they . may loses . . . be accepted as the average uncertainty of the time of a coincidence. i" in 358*, the corresponding average uncertainty of a comparison is COMPARISON BY COINCIDENT BEATS. No. Since the clock o'ooS. IHtlilZON AND VERXIEli 59 chronometer, so far as possible, should be kept in a fixed position with respect to the points of the compass. THE ARTIFICIAL HORIZON Description and use. 36. The artificial horizon consists of a shallow dish The with mercury. force of gravity brings the surface to a horizontal position, and the high reflective power of the metal makes it possible to see the various celestial bodies reflected in the surface. Any given object and its image will be filled situated on the same vertical circle, and the angular distance of the image below the surface will be equal to that of the object above. The angular distance between the object and its image is therefore twice its apparent altitude. Strictly speaking, this is true only when the eye of the observer is at the surface of the mercury, but for distant objects the error is insensible. The measurement of the distance between the object and its image therefore affords a means of determining the altitude of a celestial body, and in this connection the artificial horizon is a valuable accessory to the sextant. It can also be used to advantage with the engineer's transit for the elimination of certain instrumental errors. The artificial horizon is usually provided with a glass roof to protect the surface of the mercury from disturbances by air currents. It is important that the plates of glass should be carefully selected in order that the light rays traversing them may not be deflected from their course. The effect of any non-parallelism of the surfaces may be eliminated by making an equal number of settings with the roof in the direct and reversed position, reversal being accomplished ing the roof end for end. by turn- THE VERNIER 37. The vernier is a short graduated plate attached Description and theory. to scales for the purpose of reducing the uncertainty of measurement. It takes its name from and use. In its its inventor, Pierre Vernier, who in 1631 described its construction usual form the graduations are such that the total number of vernier divisions, which we may denote by n, is equal to i divisions of the scale, the graduation nearest the zero of the scale marking the zero of the vernier. The vernier slides along the scale, the arrangement being such that the angle, or length, to be measured corresponds to the distance between the zeros of the scale and of the vernier. When the zero of the vernier stands opposite a graduation of the scale, the desired reading is given directly by the scale. Usually this not occur, and the vernier is then used to measure the fractional part of the scale division included between the last preceding scale graduation and the zero will of the vernier. The difference between the values of a scale and a vernier division the least reading = / of the vernier. d d' If = value of one division of the = value of one division of the vernier, scale, is called PRACTICAL ASTRONOMY 60 then, for the method of graduation described above, i (n ) d = nd' whence '-*-*-; The least reading of the vernier Now, is (68) therefore i/tith of the value of a scale division. for an arbitrary setting of the vernier, consider the intervals between the various vernier graduations and the nearest preceding graduations of the scale, beginning with the zero of the vernier and proceeding in order in the direction of increasing readings. The first interval is the one whose magnitude is to be determined by the vernier. Denote its value by i'. Since a vernier division is less than a scale division by the least reading, I, it follows that the interval between the second pair of graduations will be v 2/; and so /; that between the third v each successive interval /. far on, decreasing by By proceeding enough we shall find a pair for which the interval differs from zero by an amount equal to, or less than 1/2, a quantity so small that the graduations will nearly, if not quite, Suppose this pair to be n' divisions from the zero of the vernier. The coincide. value of the corresponding interval will be v p In practice we disregard s = To V+*. therefore find (69) and use v divisions '/=, and we determine the value of v, = n'l. therefore, (70) we count the number of vernier from the zero of the vernier to the vernier graduation which most nearly coincides with a graduation of the scale. The product of this number into the reading is the value of v. The final result is the sum of v and the reading least corresponding to the last scale graduation preceding the zero of the vernier. In practice the actual counting of the number of divisions between the zero of the vernier and the coincident pair is avoided by making use of the numbers stamped on the vernier. These give directly the values of n'l corresponding to certain equidistant divisions of the vernier. Usually one or two divisions precede the zero and follow the last numbered graduation of the vernier. These do not form a part of the n divisions of the vernier, and are therefore to be disregarded in the determination of /. They are added to assist in the selection of the coincident pair when coincidence occurs near the end of the vernier. 38. Uncertainty of the result. constructed vernier is whose fectly the of result is therefore tainty 1/2. , The The error of a reading made with a perThe uncerabsolute value is 1/2. maximum gain in precision resulting from the use of the vernier may be found by comparing the uncertainty of its readings with that arising when the scale alone is used. The latter may be fixed at O.osd, as experience shows that this UNC&RTAINTT OP VEKXIER HEADINGS 61 approximately the uncertainty of a careful eye estimate of the magnitude of r. The inverse ratio of the two uncertainties may be taken as a measure of the is increase in precision, whence we find that the result given by the vernier is approx- imately ;//io times as precise as that derived from an estimate of the fractional parts of a scale division. It appears, therefore, that a vernier is of no advantage unless the number of its divisions is in excess of ten. The use of a magnifying lens usually shows that none of the vernier graduations exactly coincides with a graduation of the scale. With a carefully graduated instrument, it is possible, by estimating the magnitude of e, to push the limit given above. To do so it is only necessary with the interval between the next following pair of graduations, or with that of the pair immediately preceding, according as is positive or precision to somewhat beyond the compare e The sum of the two intervals to be compared is negative. possible to estimate s in fractional parts of the least reading. The condition that divisions of the vernier equal ;i scale must be rigorously fulfilled if reliable results /. I It is therefore divisions of the are to be obtained. The matter should be tested for different parts of the scale by bringing the zero of the vernier into coincidence with a scale graduation, and then examining whether the r ) st (+ vernier graduation stands exactly opposite graduation of the scale. Information may thus be obtained as to the accuracy with which the graduation of the instru- ment has been performed. The lie, preferably, in the same plane as the scale, and, in snugly against the latter. In many instruments, however, it rests on top, the plate being beveled to a knife edge where it touches the scale. With this arrangement the greatest care must be exercised in reading to keep all vernier should positions, should fit the line of sight perpendicular to the scale. will affect the result. Otherwise an error due to parallax THE ENGINEER'S TRANSIT 39. for the Historical. The combination of a horizontal measurement of azimuth and altitude is known circle with a vertical arc to have been used by the Persian astronomers at Meraga in the thirteenth century, and it is possible that a similar contrivance was employed by the Arabs at an even earlier date. The principle involved did not appear in western Europe, however, until the latter half of the sixteenth century. There it found its first extensive application in the who constructed a number of "azimuth-quadrants" famous observatory on the island of Hveen. The vertical arcs of Tycho's instruments were movable about the axis of the horizontal circle, and were provided with index arms fitted with sights for making the pointings. The adjustment for level was accomplished by means of a plumb line, the spirit level not yet having been invented. Magnification of the object was impossible, as a quarter of a century was still to elapse before the construction of the first telescope. The instruments were large and necessarily fixed in position and, indeed, there was no need for moving them from place to place as they were intended solely for astronomical observations. Though primitive in design, they were constructed instruments of Tycho Brahe, for his ; PRACTICAL ASTRONOMY 62 with the greatest care, and were capable of determining angular distances with an uncertainty of only i' or 2'. They are of interest not only on account of the re- markable series of results they yielded in the hands of Tycho, but also because modern altazimuth, the universal instrument, the theodolite, the engineer's transit, and a variety of other instruments. None of these modern instruments is the invention of any single person, but rather a combination of inventions by various individuals at different times. The telescope, first constructed during the early years of the seventeenth century, was they embody the essential principle of the adapted to sighting purposes through the introduction of the reticle by Gascoigne, Auzout, and Picard. Slow motions were introduced by Hevelius. The vernier was invented in 1631, and the spirit level, by Thevenot, in 1660. All these were combined with the principle of the early azimuth-quadrant to form the altazimuth, which appears first to have been made in a portable form by John Sisson, an Englishman, about the middle of the eighteenth century. At the beginning of the nineteenth century the design and construction were greatly improved by Reichenbach, who also added the movable horizontal circle, thus making it possible The universal instrument was to measure angles by the method of repetitions. then practically complete, and the transition to the engineer's transit required only the addition of the compass and such minor modification as would meet the requirements of precision and portability fixed by modern engineering practice. a detailed description of the engineer's transit, the student is referred standard work on surveying. Certain attachments, notably the compass any and the telescope level, are not required for the determination of latitude, time, For to and azimuth. On the other hand, it is desirable that the instrument used in the solution of these problems should possess features not always present in the modern instrument. In particular, the vertical circle should be complete, and should A be provided with two verniers situated 180 apart. diagonal prism for the observation of objects near the zenith, and shade glasses for use in solar observations are a convenience, though not an absolute necessity. 40. and adjustment. It is assumed methods by which the engineer's transit may Influence of imperfections of construction that the student is familiar with the be adjusted, and that observations will not be undertaken until the various adjustments have been made with all possible care. But since an instrument is never perfect, it becomes of importance to determine the influence of the residual errors in construction and adjustment, and to establish precepts for the arrangement of the observing program such that this influence may be reduced to a minimum. In the instrument fulfilling the ideal of construction and adjustment, the fol- lowing conditions, among others, are satisfied 1. The rotation axes of the horizontal : circle and the alidade coincide. planes of the circles are perpendicular to the corresponding axes of rotation. 2. The 3. The centers of the circles lie in the corresponding axes of rotation, and the lines joining the zeros of the verniers pass through the axes. 1 /.V.SV/iY.M/A'.V X 4. The 5. The The vertical axis of rotation is . 1 A BRRO/tS truly vertical when 63 the plate bubbles are centered. 6. horizontal rotation axis perpendicular to the vertical axis. is line of sight, i.e. the line through the optical center of the objective and the middle intersection of the threads, is perpendicular to the hori- zontal axis. 7. It is The when the line of sight is horizontal. the instrument maker to see that the first three of these conThe observer, on the other hand, is responsible for the re- vertical circle reads zero the task of ditions are satisfied. mainder. No. is I method of of importance only in the measurement of horizontal angles by the The error arising in such measures from non-coincidence repetitions. of the vertical axes No. 2. circles to the It may can be shown that the error due to lack of perpendicularity of the In well conis of the order of the square of the deviation. axes structed instruments No. 3. be eliminated by the arrangement of the observing pro- in Section 47. gram described it is If the third therefore insensible. condition is not satisfied the readings will be affected by an error called eccentricity. Fig. 7. In Fig. 7 let C be the center of the graduated circle OV^^ a, the point where the rotation axis intersects the plane of the circle; O. the zero of the The distance aC=e graduations; and F t and V'> the< zeros of the verniers. is the eccentricity of the circle. The perpendicular distance of a from the The reading of F, line joining V and V is the eccentricity of the verniers. is the angle OCF,, and of F,, OCV Denote these by R, and /?,, respectThe angles through which the instrument must, be rotated in order ively. that the zeros of the verniers may move from O to the positions indicated, are / =A and OaV,,=A.,, respectively. A and A., are therefore to be regarded ; ' l . l 1 l PRACTICAL ASTRONOMY 64 as the angles which determine the positions of the verniers with respect to and 2 with the vernier for the pointing in question. The relations connecting t A R readings, l and R s, are A,=R + /*, = *, + t where and Vz u and , . The mean of (71) and (72) shown telescope, we have , (73) the analogous equation = #(*/+*.') +. + # (.' and of the verniers and that ,'). (74) ,'/ 2 where are of the order of tt"/r r the radius of the circle. The last terms , the eccentricity f (73) ar>d (74) are entirely insensible in a well constructed instrument. difference of (73) and (74) is therefore e' is XW+A;) - ^(A +A = #(/?,+*.) - #(*.+*,). t V^ is t A.') easily (72) + A,) = % (R + R.) + E. + % (E, - E,). For any other pointing of the is . are tne corrections for eccentricity for the points O, 2 y> (A, It A The (75) t) The left member of (75) is the angular distance through which the instrument is rotated in passing from the first position to the second, and the equation shows that this angle is equal to the difference in the means of the vernier readings for the final and initial positions. The eccentricity is therefore eliminated by combining the means of the readings of both verniers. It can be shown that the eccentricity will also be eliminated by combining the means of any number of verniers, greater than two, uniformly distributed In practice it is sufficient to use the degrees indicated by the circle. vernier with the means of the minutes and seconds of the two readings. Nos. 4 7. Horizontal Angles: In the measurement of horizontal angles about the first an error of adjustment in No. 7 has no influence. residual errors in Nos. 4 6, let To investigate the effect of *=inclination of the vertical axis to the true vertical, 90 90 The . -f- y=inclination of the horizontal axis to the vertical axis, ^^inclination of the horizontal axis to the horizon plane, r=inclination of the line of sight to the horizontal axis. quantities b and c are the errors in level and collimation, respectively. which represents a projection of the celestial sphere on the plane Then, of the horizon, let Z be the zenith, Z' the intersection with the celestial sphere of the vertical axis produced, O an object whose zenith distance is and A in Fig. 8, , the intersection of the horizontal axis produced with the celestial sphere O is seen at the intersection of the threads. The sides of the triangles when ZAZ' INSTRUMENTAL ERRORS and ZAO have the values indicated in the figure. 65 K Finally, let k, and / be the ZA, ZO, and Z'A, respectively, referred to ZP. Applying equations (13) and (15) to triangle ZAZ\ we find directions of = sin/cos = cos/ sin sin b cos b sin / -f cos/sin /cos/, (76) (77) /. In a carefully adjusted instrument i, j, and b are very small, and we may neglect their squares as insensible. Equations (76) and (77) thus reduce to (78) (79) Equation (13) applied to triangle sin c = sin K + k are cos za ZAO + cos gives sin za cos (AT (80) ' Since c and 90 c also very small, equation (80) = b cos 2 -f (90 may be written K + k) sin z a or K 90 k = b cot z + c cosec z a . (81) no errors of adjustment, the direction of A referred to P The direction given by the instrument, determined by the angle through which it must be rotated to bring A from coincidence with ZP Were there would be AT to 90. its actual position, is /. spect to A, the difference 5 Since the verniers maintain a fixed position with re/ 90 represents the effect of the residual errors K PRACTICAL ASTRONOMY 66 But by (79) l=k, sensibly, whence it follows given by (81). If, therefore, R be the actual horizontal circle reading, and R ,, the value for a perfectly adjusted instrument. we have on the horizontal that the circle readings. amount of the error is { R, = R + 6cots + ccosecs Q C. R. , (82) Assuming that equation given by equation (78). (82) refers to that position of the instrument for which the vertical circle is on the right as the observer stands facing the eyepiece (C. R.), we find by a in which the value of b is precisely similar investigation for circle left (C. L.), b,=j z'cos/, (83) C. L. a, R the circle reading less 180, and axis to the plane of the horizon for C. L. where t is ,, (84) the inclination of the horizontal The mean (82) and of equations is (84) *,) or, substituting the values of b and b, cot*., (85) from (78) and (83) R = %(R+R + zcos/cot2 o t ) (86) . It therefore appears that the mean of the readings of the horizontal circle taken C. R. and C. L. for settings on any object is free from the influence of j, c, and the component of i in the direction of the line of sight, viz., sin /.. More- over, for objects near the horizon the effect of i cos /, the component of i parallel to the horizontal axis, is small, for it appears in (86) multiplied by cot ". If the instrument be provided with a striding level, the values of b and b l Their substitution into (85) will then give the horizontal circle reading completely freed from i, j, and cThe readings may also be freed from the influence of b by combining the results of a setting on O with those obtained by pointing on the image of seen may be determined by observation. made in the same position reflected image, O', will be on the is above. Since the the horizon as reflected in a dish of mercury, both observations being of the instrument, either C. R. or C. L. vertical circle The through O, and as far below horizontal axis is not truly horizontal, it will be necessary to rotate the instruwill thus down to 0'. slightly about the vertical axis in turning from a small amount to a new position A'. To investigate the effect of the errors for a pointing on O' we must therefore A ment move consider the triangle ZA' ^is A'ZO' in of AZO in 8. The place Fig. = ZA = gob, A'0' = A0 = go+c, and ZO' = 180 K k' where fe' is larly to equation (80), the direction of ZA' referred to ZP. A'ZO' are The angle at sides of s We . then find, simi- INSTRUMENTAL ERRORS sin c bcos 5 sin <>7 cos b sin * cos -)- (K '), whence K and, he the horizontal finally, if /?' ^ cot sa k' 90 circle the =y 2 same method we ( /? Their mean =A + J A' ' l\ , in which /?,' is cot (87) C. R. (88) reflected observation. C. c the circle reading less 180 C. R. is a, from the find . + c cosec z A") -|- , reading for the setting on O' Equations (82) and (87) both refer to C. R. By c cosec sa R'b cot z + c cosec s A> A'o -\- cosec s C. L. , L., (89) This equation combined for C. L. with (84) gives .R, = (A + A - f cosec s 5 1 14 , ,') C. L. a, (90) Equations (88) and (90) show that the mean of the horizontal circle readings for direct and reflected observations of an object in the same position of the instrument is free from the influence of any adjustment error in level. Finally the combination of (88) A' in other words the of the instrument, mean is and (90) gives =K(/?+ + A, + AY), K' (90 of the readings, direct and reflected, for both positions from b, but from the collimation error as well. free .not only Vertical Circle Readings To investigate the influence of i. j and c upon the readings of the vertical circle, consider again Fig. 8. The true zenith distance of is z; that given by the vertical circle readings is equal to the From the triangle we find angle Z'AO. : Z0 = ZAO cos sa = sin /;sin c + cos b cos c cos (ZAO). The squares and products of the errors of adjustment sible, whence we find with all necessary precision. are ordinarily quite insen- .?= Angle ZAO, Denoting the instrumental zenith distance triangle /.AZ' Z'AO by s, ZAZ', and from cos 6s\n(ca ) = sin /'sin /, we find =angle- PRACTICAL ASTRONOMY' 68 or, since c b, 2, and i are very small, z A = z + /sin (92) similar investigation gives for the reversed position of the instrument = + ?' , in C. R. /, which s, is C. L. sin/, (93) the instrumental zenith distance for C. L. The are not read directly from the circles. The ordinary angles s and engineer's transit reads altitudes, but if there is any deviation from the condition expressed in No. 7, the readings will not be the true altitudes, for they will , include the effect of the index error. and If r rt for C. R. and C. L., respectively, and / the reading izontal, be the vertical circle readings when the line of sight is hor- we have z = 90 r + /, C. R. (94) ^=90-r,-/, C. L. (95) /, C. R. (96) /-fz'sin/, C. L. (97) Substituting (94) and (95) into (92) and (93) ,; o z The mean of (96) = 90 = go and (97) s r + / + /sin 1\ is = 90 iX(r + r,)-Hsin/. (98) For an instrument whose vertical circle is graduated continuously from o it is easily shown that the equation corresponding to (98) is to 360 = in ! which v 1 t and v 3 are the % (v, ',) + / si n (99) /, circle readings, the subscripts being assigned so that <i8o*. It therefore appears that the vertical circle readings are not sensibly affected by /, c, or the component of / parallel to the horizontal axis. The component of i in the direction of the line of sight, viz., i sin/ enters with i"ts full and (98) and (99) show that it cannot be eliminated even when readings taken C. R. and C. L. are combined. The formation of the mean for the two value, positions of the instrument does eliminate the index error, however, residual error of adjustment in No. 7. To free the results from z'sin/ (i 80 za ) = the we may combine observations direct and A'ZO' previously reflected, using the mercurial horizon. Considering the triangle defined, we find for the reflected observation cos i.e. sin bs\n c + cos b cos c cos {ZA'O'} INSTRUMENTAL ERRORS 69 whence, neglecting products and squares of the errors of adjustment, the true 2 zenith distance of O' is 180 angle ZA'O'. Denoting the instrumental = zenith distance of 0', which Angle the In ZZ' = triangle the angle Z'A'O', by 180 is ZA 'Z' = (\ 80 2,) ( ZA'Z' the sides are 1 80 ZA'=go and denoting the angle ZZ'A' by /'we t, cos b sin z (z' ) s') = sin = s' 2' we s find . a Z'A'-=go b, _/, and find is'in /', or with sufficient approximation za Now cle if z' C. R. isinl', (ioo) be the vertical circle reading for C. R., reflected, and / the cir- r' reading when the line of sight is horizontal, we shall have, similarly to (94), C. R. (101) C. R. (102) This substituted into (ioo) gives from / by a quantity of the order of the errors, the difference sin I' will be insensible, so that when equations (96) and and to form the mean we have simply are combined (102) Since /' between differs i sin / j s. Similar considerations for 5U = 90X(r+r'). observations = 90 %(r, direct + r/). C. R. and (103) C. reflected, C. L. L., give (104) In other words, the formation of the means of the vertical circle readings for observations direct and reflected in the same position of the instrument eliminates not only the component of i in the direction of the line of sight, but the So index correction as well. The influence of t'cos/, j and c is insensible. far as the errors here considered are concerned, observations direct in a single position of the instrument are sufficient. Nevertheless and it is reflected desirable R that measures be made both C. and C. L. for in this way different parts of the vertical circle are used, thus partially neutralizing errors of graduation. For an instrument with a vertical circle graduated continuously from o to it is easily shown as before that in (103) and (104) the sum of the circle readings must be replaced by their difference taken in such a way that it is less than 180. 360, PRACTICAL ASTRONOMY 70 The precedirfg discussion assumes that the adjustments of the instrument remain unchanged throughout the observations. If this is not so, the elimination of the errors will, in general, be incomplete. It is not always convenient to make use of the artificial horizon, and it is method of elimination which does not therefore desirable to be able to apply a this accessory. depend upon It is easily shown that if the instrument be rclevclled before observing in the reversed position, the mean of the readings C. R. and C. L., both for the horizontal and the vertical circle, will be free from the errors in all of the ad7, within quantities of the order of the products and The same will be true, even though the plate bubbles squares of the errors. are not accurately centered during the direct observations, provided, after reversal, they be brought to the same position in the tubes that they occupied justment under Nos. 4 before. That such will be the case follows from a consideration of Fig. 8. The reversal and relevelling is equivalent to rotating the triangle ZAZ' about Zthrough the angle i8o+2c, its dimensions remaining unchanged. thus assumes a A A lt distant from position >by 90+^, and Z' a position Z t '. The triangle ZA^O leads to an equation differing from (84) only in that b t is replaced by b. The mean of the new equation and (82) is simply new R.= %(R + R,), (105) where R and R 1 are the horizontal circle readings; the latter having been reduced by 180. The result is therefore free from both b and c. Again, from triangle ZA^Z^, we find for circle left analogously to (97), z = 90 I r^ z'sin/, C. L. (106) which the vertical circle reading r is not the same as the r l of (97), for (106) presupposes that the instrument is relevelled after reversal, while (97) assumes that no change is made in the position of the vertical axis during the observations. The mean of (96) and (106) is in 1 ^0 which is free from b, c, and /. = 90 l A(r-\- For a circle r,), (107) graduated continuously we have similarly, 2,= where ference y2(v l -i',) (1 08) as before the readings are to be taken in such an order that their difis It is less than 180. assumed throughout that the pointings are always made by bringing the object accurately to the intersection of the threads. It is important that this be done, even though the threads be respectively horizontal and vertical; for observing at one side of the field is equivalent to introducing an abnormal value INSTRUMENTAL ERRORS THE LEVEL 71 of the collimation, while pointings above or below the horizontal thread correspond to a modification of the index error of the vertical circle. 41. Summary summarized The preceding of the preceding section. results may be as follows: I. Non-coincidence of vertical axes enters only when the horizontal used by the method of repetitions. Error eliminated by proper arrangement of observing program. See Section 47. No. 2. Non-perpendicularity of circles to axes usually has no sensible No. circle is influence on circle readings. No. 3. Eccentricity of circles and verniers eliminated by forming means of readings of both verniers. See equation (75). Nos. 4 7. Horizontal circle readings: Component of deviation of vertical axis from vertical in direction of line of sight, non-perpendicularity of axes, and collimation eliminated by forming mean of readings taken C. R. and C. L. Component of deviation from vertical which is parallel to horizontal axis appears mulSee equation (86). Correction for the latter may be made by tiplied by cots . observations with the striding level. See equation 6 (85). All errors in Nos. 4 eliminated by forming mean of readings direct and reflected, for both C. R. and C. L. See Equation (91). The error in No. / index error of vertical circle does not enter. Vertical circle readings: All errors ponent of deviation of vertical axis from vertical insensible or eliminated from mean in Nos. 4 7 excepting com- direction of line of sight of readings C. R. and C. L. See equation in 7 insensible or eliminated from mean of See equations readings, direct and reflected, in same position of instrument. and Desirable to observe R. both and C. to reduce C. (104). however, (103) L., (98) or (99). All errors in Nos. 4 graduation error of vertical All errors under Nos. 4 circle. 7 eliminated from mean of readings C. R. and C. L. for both horizontal and vertical angles provided plate bubbles have same See equations (105) position in tubes for both positions of the instrument. and (107) or (108). 42- The level. The adjustment of the engineer's transit with respect to the vertical is usually made by means of the plate bubbles, any residual error being eliminated by some one of the methods of Section 40. In some cases, however, it is desirable to remove the effect of this error by measuring the inclination of the horizontal axis to the horizon and applying a suitable cor- rection to the circle readings. This method of procedure requires a knowledge of the theory of the striding level. The striding level is more sensitive than the plate bubbles, its tube is It is made in two longer, and the scale includes a larger number of divisions. forms, one with the zero of the scale at the middle of the tube, the other with Theoretically the two forms are equivalent. The adjustits mounting should be such that the bubble stands at the middle of the tube when the base line is horizontal. The scale reading of the middle of the bubble for this position is called the horizontal the zero at the end. ment of the level tube within reading. Owing to residual errors of adjustment, the horizontal reading will PRACTICAL ASTRO NOMT 7J not usually be zero, even for the form in which the zero of the scale is at the middle of the tube. Its value must be determined and applied as a correcThe latter tion to the scale readings, or else its influence must be eliminated. and the direct the in made is by combining readings accomplished easily reversed position, reversal being -Let made by turning the level end for end. d= the angular value of one h = the horizontal reading. division of the level scale. be the readings of Further, for any inclination of the base line, let m' and m" b" the b' and and of the bubble, the middle corresponding observed inclinadirect and level Finally, assume that all for the reversed, respectively. tions, all toward the left, negand are the toward right positive, readings increasing of the scale, of the zero the whatever then ative. find, position We b'=(m'h)d, (109) b" (i 10) = (m" K)d. Since h has opposite signs for the two positions of the level, the and (no) mean of (109) is b=y (m' + m")d, 2 (in) which the mean of the observed inclinations has been written equal to b. Denoting by r', /', and r", /", the readings of the ends of the bubble for two positions, and writing in D=y 4 d, we find (112) from (in) ("3) This result depends only upon the readings of the ends of the bubble and the value of one division of the scale, and is therefore free from the horizontal reading. The convention regarding the algebraic sign is such that when b calculated from (113) is positive, the right end of the level is high. Since b' and b" are two observed values of the same quantity, we find from the difference of (109) and (no) h=^(r' +/' /'/"), (114) which may be used for the calculation of h when a complete observation has been made. Precepts for the use of the striding level. The level is a senand great care must be exercised in its manipulation if precise results are to be obtained. The inclinations to be measured should be small and the horizontal reading should correspond as closely as possible with the scale reading of the middle of the tube. The points of contact of the level with the pivots upon which it rests must be carefully freed from dust particles. 43. sitive instrument, THE LEVEL 73 of the bubble, which is adjustable in the more sensitive forms, should be about one-third the length of the tube, and ample time should be allowed for the bubble to come to rest before reading. The instrument should be protected from changes in temperature, and, to this end, it should be The length shielded from the rays of the sun, and from the heat of the reading lamp and the person of the observer. The right end of the bubble should always be read first, careful attention being given to the algebraic sign, reversal for each observation should be noted. avoided by noting that The in following, Mistakes in and the time of reading may /". /', the length of the bubble, must equal r" which 6' represents the sum of the four readings, be r' is a convenient form for the record: Time r r' r" S r" r' +/" r>' 4- /' "I 6 -S, = Sd. \ most easily found by forming first the diagonal sums of the four readings written as above, for both r' and I" and r" and /' will be opposite in sign and approximately equal in absolute magnitude. is The following illustrates the record and reduction of level observations". made with a level whose zero point is in the middle of the tube; the are 8'.'i6 and 0*032, respectively. second, with one whose zero is at the end. The values of Example The first 28. observation was D 0=6 h + 14.1 + io.\ + T=9 b 15'" - 9.7 +31-0 13.8 20.3 I2 m + 16.4 35.0 ; The ob44. Determination of the value of one division of a level. server should be familiar with the sensitiveness of all the levels of his instrument, even though he depends entirely upon a simple centering of the bubble If the striding level is to be used, a knowledge of the anfor the adjustment. of of its scale is an essential. value one division gular The investigation of levels is most easily carried out with the aid of a level trier, which is an instrument consisting essentially of a rigid base carrying a movable arm whose inclination to the horizon may be varied by a known amount by means of a graduated micrometer screw. The entire transit may be mounted on the arm, or the various levels may be attached separately for the investigation. The determination of the change in the inclination of the arm of the level trier necessary to move the bubble over a given number of divisions gives at once the angular value of one division of the scale. PRACTICAL ASTRONOMY 74 29. The following shows part of the reduction of observations made with a determination of the value of one division of a level. The bubble was run from the left to the right end of the tube and back again, for both level direct and reversed, the micrometer head through four divisions at a time. The ends of the bubble Example level trier for the by moving were read for each setting of the micrometer. Column two of the table gives the micrometer means of the end readings of the settings; and columns three and four, the corresponding bubble for level direct. The fifth column contains the means of the quantities in the two preceding columns; and column six, the differences between the th and the (6-f)th readings column five. The principle used in combining the observations is the same as that emmicrometer screw ployed in Examples 26 and 27. The length of the arm and the pitch of the are such that a rotation of the micrometer head through one division changes the inclination by i". Each of the displacements of the bubble in column six therefore corresponds to a in in inclination of 24". The quotients formed by dividing the displacements into 24" similar reducare the values of one division of the level for different portions of the tube. tion of the readings taken with the level in the reversed position gave for d the values in column change A eight. The means column is for the two series are in the last column. sufficient to show ONE No. A that the curvature of the level tube DIVISION OF A LEVEL glance at the results in this is variable. LEVEL TRIER ONE DIVISION OF THE LEVEL of the horizontal circle, whence the angular value 75 of one division of the level may be determined as before. To express d as a function of i and the horizontal circle readings, let HC and in Fig. 9 represent portions of the horizontal circles for the normal and the deflected positions of the vertical axis; L, any position of the level, which is supposed to be attached with its axis perpendicular to the radius through L and parallel to the plane of the circle; and b, the corresponding HC inclination. HLC In the spherical right triangle the angle //is equal to i, the Fig. 9 b. deflection of the axis from the vertical, while that at L is 90 Now, if ra and r be the horizontal circle readings corresponding to the inclinations zero and b respectively, we find Arc//=90 whence from the triangle (r r,), HLC, = tan tan b rc ). i sin (r (US) The angle b is very small and, for i equal two or three degrees, r ra will never exceed one degree. We may therefore use the approximate relation = (r with an error not exceeding o"oi. For any other inclination, ,, bi r tan ) we have (r l (116) the analogous equation ra) tan i, tan i. which, combined with (112) gives bi The angle r ding to the equal to sd, l change where s r is b r) (117) the change in the horizontal circle reading corresponThe latter, however, may be written b. in inclination is = (rt , the displacement of the bubble in scale divisions, and the angular value of one division. thus have finally as the expression for We r, d r tan /. (118) d PRACTICAL ASTRONOMT should be two or three degrees for the investigation of the If the ordinary transit levels. For very sensitive levels it should be less. of the vertical the deflection instrument be provided with a telescope level, axis may be accomplished as follows: Level the instrument and center the The angle i the vertical circle reading by the angle i and, levelling screws, bring the telescope bubble back to the tube, taking care at the same time that the transverse plate telescope bubble. by means of the Then change middle of its bubble is also centered after the deflection. This precaution is necessary in order that the deflection may have no component perpendicular to the plane In the absence of a telescope level, level the instrument, sight on a distant object, change the vertical circle reading by i, and bring the object back to the intersection of the threads by means of the levelling screws. of the vertical circle. The observations may be made either by displacing the bubble through a definite number of divisions and noting the corresponding change in the horizontal circle readings, or by changing the circle readings by a definite amount, For short say 10', and observing the variations in the position of the bubble. tubes with only a few graduations the former method is more convenient, while the latter is to be preferred for the long finely graduated tubes of sensitive levels. The bubble should be run from one end of the tube to the other and then back again, in both positions of the instrument. Such a series of readings constitutes a set. The instrument must be as rigidly mounted as possible, preferably on a desirable to check the constancy of i by deflecting through masonry pier. this angle toward the vertical at the end of a set and noting whether the It is instrument is Example then levelled. 30. Observations were made by the deflected axis method for the determina- tion of the value of one division of the striding level of a Berger transit. The deflection was 3. The graduations of the tube are in two groups of three each, the groups being separated ONE DIVISION OF Level Divisions A LEVEL DEFLECTED Axis MEASUREMENT OF ALTITUDE 77 by a space approximately equal to the length of the bubble. The horizontal circle was read when the bubble was symmetrically placed with respect to the pairs of graduations indicated The circle readings themselves are in columns two and three; in column one of the table. and the minutes of the means of corresponding settings, in column four. The differences of the readings for a displacement of the bubble through two divisions are in the fifth column. The calculation for the determination of d is in accordance with equation (118). The measurement of vertical angles. The observer will have 45. occasion to measure the altitude not only of rapidly moving equatorial stars but also of circumpolar objects like Polaris whose positions with respect to the horizon change but slowly. The difference in motion in the two cases necessitates a difference of method in making the settings. For Polaris or any other close circumpolar object, the star should be brought to the intersection of the threads by the slow motions, the time of coincidence and the vertical circle readings being carefully noted. For stars whose altitude varies rapidly, thrs cannot be done with precision. The object is therefore brought into coincidence with the vertical thread near the point of intersection, and kept on the thread by slowly turning the horizontal slow motion until the instant of transit across the horizontal thread, the time and the vertical circle readings being noted as before. Observations on the sun are most readily made with the aid of a shade of colored glass, but if this is not available, the image may be projected on a card held a few inches back of the eyepiece, by a proper focusing of the objective. In order that the threads may be seen sharply defined on the card, it is necessary that the eyepiece be drawn out a small fraction of an inch from its normal position before the solar image is focused. There are several methods by which the pointings may be made. For example, the instrument may be adjusted so that the preceding limb is near the horizontal thread and approaching the intersection. The instrument is clamped and the instant of tangency carefully noted. Then, without changing the vertical circle reading, the image is allowed to trail through the field until the transit of the following limb occurs, when the time is again noted, the instrument in the meantime being rotated by means of the horizontal slow motion so that both transits are observed at the intersection of the threads. While waiting for the second This method is open to the objection that transit, the vertical circle is read. an interval of three or four minutes separates the transits of the two limbs, which entails a considerable loss of time. The interval may be shortened by shifting the position of the telescope between the observations, but this of course requires a reading of the vertical circle for each transit. If there be more than one horizontal thread, the difficulty can be avoided by observing the transits over the extreme threads the preceding limb over the first thread and the following limb over the last thread. The same number of The mean of the readings will then settings should be made for both limbs. correspond to the altitude of the sun's center, the influence of semidiameter being eliminated. If for any reason the program cannot be made complete in this particular, the altitude of the sun's center may still be found with the PRACTICAL ASTRONOMT 78 aid of the value of the semidiameter interpolated from page for the instant of observation. I of the Epliemeris of the observing program is determined by the results Section 40 and summarized in Section 41. The number of settings to be made for the determination of the altitude depends upon the precision desired, the rapidity with which the observer can make the pointings and read The arrangement derived in the circle, and the position of the object. number should not be less than two maximum number to be included in It is desirable, however, that the The for each position of the instrument. a single set is limited by the fact that it is convenient to use for the reduction the means of the circle readings and the Since the change in the altitude of the star is not proportional to the times. the time, the two means, rigorously speaking, will not correspond to in change each other; but if the observing interval does not exceed a certain limit, say a quarter of an hour, no appreciable error will be introduced into results secured with the engineer's transit by treating the means as a single observation. The observing program will also depend on the method employed for the elimination of the instrumental errors tors involved, we adopt observations on a will at star. /,_/, c and /. Bearing in mind the various fac- the following as convenient arrangements for a set of The necessary modifications for measures on the sun once be suggested by the methods for making the settings described in the preceding paragraph. OBSERVATIONS DIRECT OBSERVATIONS DIRECT AND REFLECTED Level. Level. readings on star, C. R. Reverse 2 2 i Level. i 4 readings Reverse. on star, C. L. I 2 Readings on With the first star, C. R. arrangement, which \ \ C.R. I Reverse. Level. 2 reading on star, direct. readings on star, reflected. reading on star, direct. i is reading on star, direct. readings on star, reflected. VC.L. ) reading on star, direct. to be i used when all of the pointings made directly on the star, the elimination of the errors depends upon the bubbles occupying the same positions in their tubes for both C.R. and C.L. The instrument must therefore be relevelled carefully after each reversal. are With the second, which will find application when the artificial horizon is elimination will be complete if the adjustments remain unemployed, the intervals changed during separating the various direct observations and the corresponding reflected observations immediately preceding or following. After the instrument has once been levelled, therefore, the screws need not be touched until the set has been completed unless the bubbles should become the displaced by a considerable amount. Both verniers should be read for each setting of the telescope. If only an approximate result is required, the observations may be discontinued at the middle of the set. On the other hand, if more precision is MEASUREMENT OF ALTITUDE desired, additional sets may 79 be observed, each of which, however, should be reduced separately. The fact that for a short interval the change in the altitude is sensibly proportional to the change in the time makes it possible to test the consistency of the measures. For direct observations the quotients of the differ- ences between the successive circle readings by the differences between the corresponding times must be sensibly equal. If this condition is not satisThe errors most likely to occur are fied, an error has been committed. those involving mistakes of 10' or 20', or perhaps a whole degree, in the circle readings, and an exact number of minutes in the times. It is convenient to express differences of the circle readings in minutes of arc, and the time intervals in minutes and tenths. The quotients will thus express the change in the altitude in minutes of arc for one minute of time. If the artificial quotients must be calculated for the direct and reflected observations separately. For observations on the sun, the combination of the data for the calculation of the quotients will depend upon the horizon been has used the method followed in making the settings, and is easily derived in any special case. The test is usually sufficient to locate errors of the class mentioned with such certainty as to justify a correction of the original record, and should always be applied immediately after the completion of the set in order that the measures may be repeated if necessary. For circumpolar objects, a simple inspection will usually be sufficient to indicate the consistency of the observations. Equations (103) and (104), and (107) show that for an instrument graduated to read altitudes, the apparent altitude, free from the instrumental errors, ',_/', and /, will be given by forming the mean of the circle readings obtained in accordance with the above programs. For an instrument with its vertical circle graduated continuously o to 360 the zenith distance will be given by c, z =Yt(v t v,) (119) where the subscripts are assigned in such a manner that i>, If the v, < 180. observations are direct, one v will represent the mean of all the circle readings If the artificial horizon has been used, C.R.; the other, the mean of all C.L. will represent the mean of all mean of all the reflected readings. The observed altitude, or zenith one v for refraction and parallax in the direct readings; and the other, the distance, thus derived must be corrected accordance with Sections 8 and 9. 31. The following is the record of partial sets of observations made with a Buff engineer's transit at the Laws Observatory, on 1908, Oct. 2, Friday P. M., for the determination of the altitudes of Polaris and Alcyone. The measures were all direct. The Example Buff & timepiece used was an Elgin watch. An inspection of the readings for Polaris shows that the measures are consistent. The relatively large difference in the readings C. R. and C. L. reveals the existence of an index error of 2' or 3'. PRACTICAL ASTRONOMY 80 ALCYONE POLARIS Vertical Circle Vertical Circle Watch 19" Ver. 39 A 26' B Ver. 39 26' 25 26 26 36 50 36 ii 33 33 34 34 33 8" 35-1 1- 39 29'. 8 Clrcle Watch R R L L m Ver. A Ver. B Clrcle Rate 2i!4 20 40' 20 37 1.2 20 59 20 59 4i i-4 21 38 21 38 L R 43 1.4 22 i 22 I R 91139"' 6H 9>>3s 40' ^ _ 2ii9-5 For Alcyone the close agreement of the values for the rate of change in altitude per minute of time given in the last column is evidence of the consistency of the measures. The quantities in the fifth line are the means. The angles are the apparent altitudes obtain the true altitudes a corresponding to the watch times immediately preceding. To correction for refraction, which may be obtained from Table J, page 20, must be applied. Example 32. The following observations were made with a Berger engineer's transit on 1908, October 15, Thursday P. M., for the determination of the altitude of the sun. The measures were all direct and were made by projecting the image of the sun on a card. The transits were observed over the middle horizontal thread, the telescope being shifted after each transit. The timepiece was the Fauth sidereal clock of the Laws Observatory. Fauth HORIZONTAL ANGLES 1 DIRECT OBSERVATIONS setting on mark \ on star on star setting on mark 2 settings i 2 settings 1 I / DIRECT AND REFLECTED OBSERVATIONS DIRECT OBSERVATIONS i I 3 3 i i setting on mark setting on mark settings on star settings on star setting on mark setting on mark 81 mark C.R. mark C.L. C.R. setting on C.I.. setting on C.I,. setting C.R. C.R. C.L. setting on star, on direct star, 1 _, reflected / on star, reflected \ on star, direct J setting on mark C.R. setting on mark C.L. setting R setting Both verniers of the horizontal circle should be read for each setting, and made on the star, the time should be noted in addition. The required difference of azimuth will be the difference between the means of the readings on the mark and on the star. Its value will correspond for those mean of the times. If more precision is desired than can be obtained from a single set, several sets may be observed, each of which should be reduced separately. To reduce the influence of graduation error, the horizontal If the number of sets is n, the circle should be shifted between the sets. amount of the shift between the successive sets should be 36o/. to the Example 33. The following is the record of a simultaneous determination of the altitude of Polaris and the difference in azimuth of Polaris and a mark. ALTITUDE OF POLARIS AND AZIMUTH OF MARK No. 1908, Oct. 13, Thursday P. Station No. 2 Buff Jrw = Object Mark 2 M. Observer Sh. Recorder W. & Buff Engineer's Transit No. 5606 h 38*7 at 7 59 P.M., and 31:4 at 9" 54 P.M. PRACTICAL ASTRONOMY 82 increase the reading by the angle D, while that from B back to A will produce no change since during this rotation the vernier remains clamped to the circle. If the turning from A to B is repeated n times, the difference between the circle readings for the final setting on B and the initial setting on A will be nD; and if the initial and final readings be R, and R,, repectively, we shall have D = ^=A. The method of repetitions derives its (I20) advantage from the fact that the not read for the intermediate settings on A and B. Not only is the observer thus spared considerable labor, but, what is of more importance, the errors which necessarily would affect the readings do not enter into the result. circle is Consequently, that part of the resultant error of observation arising from the intermediate settings is due solely to the imperfect setting of the cross threads on the object. For instruments such as the engineer's transit, in which the uncertainty accompanying the reading of the angle is large as compared with that of the pointing on the object, the precision of the result given by (120) will be considerably greater than that of the mean of n separate measurements of the angle D, each of which requires two readings of the circle. But for instruments in which the accuracy of the readings is comparable with that of the pointings, as is the case with the modern theodolite provided with read- ing microscopes, the method of repetitions is not to be recommended. Although there is even here a theoretical advantage, it is offset by the fact that the peculiar observing program required for the application of the method presupposes the stability of the instrument for a relatively long interval, and hence affords an unusual opportunity for small variations in position to affect the precision of the measures. Moreover, experience has shown that there are small systematic errors dependent upon the direction of measurement, i.e. is made on A or on B; and, although these may be eliminated in part by combining series measured in opposite directions, it is not certain that the compensation is of the completeness requisite for observations of the highest precision. With the engineer's transit, however, the method of repetitions may be used with advantage. Since rotation takes place on both the upper and the lower motions, any non-parallelism of the vertical axes will affect the readings; and the observing program must be arranged to eliminate this along with the other instrumental errors. For any given setting the deviation of the axis from parallel- upon whether the initial setting ism,/, unites with the inclination of the lower axes to the true vertical, i', and determines the value of z, the inclination of the upper axis to the vertical, for the For different settings i will be different, for a setting in question. rotation of the instrument on the lower motion causes the upper axis to describe a cone whose apex angle is 2p and whose axis is inclined to the true vertical by i'. But no matter what the magnitude of i may be, within certain limits easily including all values arising in practice, it may be eliminated forming the mean of direct and reflected readings made in the same by position METHOD OF REPETITIONS 83 and magnitude for on both settings. This follows from the discussion pages 66 and 67 whose result is expressed by equation (88). Hence, if after a series of n repetitions observed C.R. direct, n further repetitions be made C.R. reflected, such that the vernier readings for the corresponding settings in the two series are approximately the same, the instrumental errors i' and /will be eliminated. Equation (88) shows that j, the deviation of the upper vertical axis from perof the instrument, provided that i is the same in direction pendicularity with the horizontal axis, will also be eliminated. To remove the influence of the collimation, c, the entire process must be repeated C.L.; to neutralize the systematic error dependent upon the direction of measurement, the direct and reflected series should be measured in opposite direcWe thus have the following observing program, in which A' and tions. and ' denote the reflected images of A and B, respectively: Level on the lower motion. (Set Direct -j on A and read the horizontal circle. Turn from A to B on the upper motion times. Read the horizontal circle for last setting on B. ^ Set on B' and read the horizontal circle. C.R. ( Reflected <. I The circle Repeat for C.L. reading for the first setting on B' must be the same, approximately at as that for the last setting The mean difference of A Turn from B' to A' on the upper motion times. Read the horizontal circle for last setting on A'. on least, />'. of the values of D calculated from the four series is the required azimuth and B. Uusually one of the objects, say A, will be near the horizon, in which case A must then be substituted for reflected settings on A' will be impossible. error to i will not be eliminated from these above The due A' in the program. of to the the factor cot2 it may be disregarded. settings; but, owing presence When the artificial horizon is not used the program must be modified. Were i' zero, i would constantly be equal to/, although the direction of the deflection would change with a rotation of the instrument on the lower motion. If a series of n repetitions C.R. be made under these circumstances, equation (82) shows that each setting will be affected by an error of the form , /cot -f- p cos /cot 2 -f- c cosec 2 . and last terms of this expression will have the same values for all on same object. Equations (82), (84), and (86) show that they may the pointings be eliminated by combining with a similar series made C.L. The values of the second term will be different for each setting owing to the change in I, but their sum will be zero if the values of / are uniformly distributed throughout 360, or any multiple of 360. In order that this may be the case, approximately at least, it is only necessary that n be the integer most nearly equaling k 36o/Z>, where the k is any integer, in practice usually I or 2. The first It is also easily seen that, if after any arbitrary number of settings the instrument be reversed about the loiver motion and the series repeated in the PRACTICAL ASTRONOMT 84 sum of the errors involving/ will be zero, provided that the and C.L. are the same, or apfor circle readings corresponding settings C.R. The reversal of the instrument on the lower motion changes so. proximately 180. The values of / for corresponding of the reverse order, the deflection/ by and C.L. will therefore differ by 180, and the errors will be oppoof the two series is formed. The site in sign and will cancel when the mean in the preceding reversal also eliminates the influence of j and c as indicated the direction settings C.R. paragraph. If that the deflection of the lower axis, z', is zero. of the additional error an be affected will each by the not setting case, this is form i' cos/' cot 2 in which /' is constant so long as i' remains unchanged If i' be the result of a non-adjustment of the plate bubbles, the in direction. The above assumes , which it produces may be eliminated from the mean of two series, one C.R. and one C.L., by relevelling after reversal. (See page 70.) This will the values of /' for C.R. change the direction off by 180. Consequently, the two for errors the and differ will positions will neutralize C.L. 180, and by error mean is formed. The consideration of these results each other when the leads to the following arrangement of the observing program. Level on the lower motion. Set on A and read horizontal circle. B on upper motion Turn from A to Read horizontal ^ times. circle for last setting \ C.R. on B. Reverse on lower motion and relevel. Set on B and read horizontal circle. Turn from B to Read horizontal A \ on upper motion times. circle for last setting on A. I C.L. > at setting on B, C.L. should be the same, approximately on B, C.R. The mean of the values of D calculated from the two series is the required azimuth difference of A and B. The circle reading for the first least, as that for the last setting arrangement the instrumental errors i',p,j, and c will be comor not, pletely eliminated, whether the settings are distributed through 360 obserthe provided only that the instrumental errors remain constant during that nD of n be such should vations. Practically, it is desirable that the value With this D is small equals 360, or a multiple of 360, at least approximately; but when of the observations. The maximum number, this may unduly prolong repeti- which can be made advantageously depends upon the stability of the instrument and must be determined by experience. If the instrument is provided with a striding level, the influence of i\ p, and / may be taken into account by measuring the inclination of the horizontal tions R R axis for each setting and applying a correction to and 3 of the form cotz in which b denotes the sum of all the observed inclinations for settings on A B respectively. When one of the objects, l , and say B, is a star, the time of each setting on B calculated value of will then correspond sensibly to of the times, provided the observing program be not too long. must be noted. the mean The D THE SEXTANT Example 34. The 1909, April 9, the following observations of the difference in azimuth of repetitions with a Buff & Buff engineer's m 36". recorded times are those of a Fauth sidereal clock whose error was of Polaris and a transit. On 85 mark were made by the method +6 After four repetitions C.R., the instrument was reversed on the lower motion, relevelled, and the series repeated in the reverse order. Since the azimuth difference is approximately 174, 720 must be added to the readings on the star before combining them with those on the mark. The results for the are also given. two halves are derived separately, although the means for the set PRACTICAL ASTRONOMY 86 and to-day the sextant is the only instrument which practice of navigation, can advantageously be employed in the observations necessary for the determination of a ship's position. In addition, its compactness and lightness, and the precision of the results that may be obtained with it render it one of the most convenient and valuable instruments at our command. The modern sextant consists of a light, flat, metal frame supporting a graduated arc, usually 70 in length; a movable index arm; two small mirrors perpendicular to the plane of the arc; and a small telescope. The index arm of the pivoted at the center of the arc and has rigidly attached to it one axis contains the rotation surface whose reflecting mirrors, the index glass, of the index correThe mirror. of the arm and the attached glass position sponding to any setting may be read from the graduated arc by means of a The second mirror, the horizon glass, is firmly attached to the vernier. is frame of the sextant in a manner such that when the vernier reads zero the two mirrors are parallel. Only that half of the horizon glass adjacent to the frame is silvered. The telescope, whose line of sight is parallel to the frame, is directed toward the horizon glass, and with it a distant object may be seen through the unsilvered portion. When the frame is brought into coincidence with the plane determined by the object, the eye of the observer, and any other object, a reflected image of the second object may be seen in the field of the telescope, simultaneously with the first, by giving the index arm depending upon the angular distance separatthe position of the arm is such that the rays of the second object reflected by the index glass to the horizon glass, and then from the silvered portion of the latter, enter the telescope parallel to the rays that pass from the first object through the unsilvered portion of the horizon glass, the a certain definite position ing the objects. two images If be seen This being the case, the relative inclibe one-half the angular distance the is such that the inclination since the construction objects; and, separating be read from the it becomes may graduated arc, possible to find the angular distance between the objects. The use of the instrument is simplified by graduating the arc so that the vernier reading is twice the inclination of the will in nation of the mirrors as coincidence. shown below, will With the mirrors, and hence, directly, the angular distance of the objects. usual form of the instrument the maximum angle that can be measured is therefore about 140. The two mirrors and the telescope are provided with adjusting screws, which may be used to bring them accurately into the positions presupposed by the theory of the instrument. In addition, the tele- scope may be moved perpendicularly back and forth with respect to the frame thus permitting an equalization of the intensity of the direct and reflected images by varying the ratio of the reflected and transmitted light that enters the telescope. Adjustable shade glasses adapt the instrument for observations on the sun. 49. The principle of the sextant In Fig. 10 let 0V represent the graduated arc; / and H, the index glass and the horizon glass, respectively; and IV, the index arm, pivoted at the center of the arc and provided with a THE SEXTANT 87 When V coincides with 0, the mirrors are parallel. The posithat the two objects 5, and S, are seen in such tion indicated from for the 5, pass through the unsilvered portion of coincidence, rays direction in the and enter the telescope HE, while those from S, falling on / vernier at V. in the figure is are reflected to H H and thence in the direction HE. The two beams therefore enter the telescope parallel. H is one-half the angular be shown that the inclination of / to are the normals to the mirrors, separating the objects. /<Vand and by the fundamental laws of reflection they bisect the angles and IHE, respectively. In the triangle IHE. It is to distance HN A SJH = + A, a = b+ Y A. 2a whence 2i> 2 But in the triangle IHN a = b + M. Therefore, But M, being the angle between the normals to the mirrors, measures their and is equal to the angle subtended by the arc 0V, whence inclination, A = 20y. (121) But since the arc by OF the is graduated so that the reading is twice the angle subtended angular distance between the two objects is given directly by the scale. 50. among Conditions fulfilled by the instrument. The following conditions, must be fulfilled by the perfectly adjusted sextant. others, PRACTICAL ASTRONOMT 88 1. 2. 3. 4. 5. The index glass must be perpendicular to the plane of the arc. The horizon glass must be perpendicular to the plane of the arc. The axis of the telescope must be parallel to the plane of the arc. The vernier must read zero when the mirrors are parallel. The center of rotation of the index arm must coincide with the center of the graduated arc. Since the positions of the mirrors and the telescope are liable to derangement, methods must be available for adjusting the instrument as perfectly as This is the more important inasmuch as it is impossible to eliminate possible. from the measures the influence of any residual errors in the adjustments. Although elimination is impossible, it should be remarked that the errors arising in connection with Nos. 4 and 5, at least, may be determined by the methods given in Sections 52 and 53, and applied as corrections to the readings obtained with the instrument. Conditions Nos. 1-4 are within the control No. 5 must be satisfied as perfectly as possible by the of the observer. manufacturer. 51. Adjustments of the sextant. No. i. Index glass. To test the perpendicularity of the index glass, place the sextant in a horizontal position, unscrew the telescope and stand it on the arc just in front of the surface of the index glass produced. If then the eye be placed close to the mirror, the observer will see the reflected image of the upright telescope alongside the telescope itself. By carefully moving the index arm, the telescope and its image may be brought nearly into coincidence. If the two are parallel, the index glass is in adjustment. The telescope should be rotated about its axis in order to be sure that it is perpendicular to the plane of the arc. If the adjustment imperfect, correction must be made by the screws at the base of the mirror. Some instruments are not provided with the necessary screws, and in such cases the adjustment had best be entrusted to an instrument is maker. The test can also be made by looking into the index glass as before, and noting whether the arc and its reflected image lie in the same place. If not, the position of the mirror must be changed until such is the case. No. 2. The horizon glass. The adjustment of the horizon glass may be by directing the telescope toward a distant, sharply defined object, preferably a star, and bringing the index arm near the zero of the scale. Two images of the object will then be seen in the field of view one formed by the tested rays transmitted by the horizon telescope by the mirrors. image as the index arm is glass, the other, by those reflected into the The reflected image should pass through the moved back and forth by the slow motion. direct If it does not, the horizon glass is not perpendicular to the plane of the arc, and must be adjusted until the direct and reflected images of the same object can be made accurately coincident. No. 3. The telescope. The parallelism of the telescope to the frame may be tested by bringing the images of two objects about 120 apart into coincidence at the edge of the field nearest the frame. Then, without changing INDEX CORRECTION 89 If they remain the reading, shift the images to the opposite side of the field. in If its is the in coincidence, not, adjustment. position must be telescope collar of the screws of the until the test means varied by adjusting supporting is satisfactory. No. 4. Index adjustment. If the fourth condition is not fulfilled, an index error will be introduced into the angles read from the scale. To test the adjustment, bring the direct and reflected images of the same distant object The corresinto the coincidence as in the adjustment of the horizon glass. R If is zero, the ponding scale reading is called the zero reading adjustment is correct. If not, set the index at o, and bring the images into coincidence by means of the proper adjusting screws attached to the horizon It is better, however, to disregard this adjustment and correct the readings glass. by the amount of the index error. It can be shown that the errors affecting the readings as a result of an imperfect adjustment of the index glass, the horizon glass, and the telescope are of the order of the squares of the residual errors of adjustment. If care be exercised in making the adjustments, the resulting errors will be negligible as compared with the uncertainty in the readings arising from other sources. = . R Make a series of zero 52. Determination of the index correction. readings on a distant, sharply defined object, a star if possible. If the zero of the vernier falls to the right of the zero of the scale, do not use negative readings, but consider the last degree graduation preceding the zero of the and read in the direction of increasing graduations. The zero scale as 359, reading is what the instrument actually reads when it should read zero. The index correction, /, is the quantity which must be added algebraically to the scale readings to obtain the true reading. We therefore have 7=0 ^, 7=360 (122) R (123) a. The latter expression is to be used for the determination of 7 when the zero of the vernier falls to the right of that of the scale for coincidence of the direct and reflected images of the same object. When observations are to be made on the sun, the index correction should be determined from measures on this object. Since it is impossible, on account of their size, to bring the solar images accurately into coincidence, we determine the zero reading as follows: Make the two images externally tangent, the reflected being above the direct, and read the vernier. the mean of a series of such readings. Then make an equal Let R, represent number of settings the mean of the corres- for tangency with the reflected image below. Call ponding readings R,. The mean of R, and R, will then be the value of the zero reading, and we shall have 7=o -#(*,+*.), 7 =360-^ (/?, + *,). (124) (125) PRACTICAL ASTRONOMY 90 thus obtained will also give the value of 5, the sun's semia distance of diameter. Since the center of the reflected image moves over we have the to first second, the from position four semi-diameters in shifting The readings to the brilliancy of the solar image, its diameter appears larger of than it really is a phenomenon known as irradiation. Should the value Section sun the on observations of reduction (see 55),, for the S be required Owing the value calculated .from equation (126) should be used rather than that derived from the Ephemeris, in order that the influence of irradiation may be eliminated. in the 53. Determination of eccentricity corrections. Any defect with into the Since, error an introduces readings. condition fifth eccentricity the usual form of the instrument there is but a single vernier, this cannot be eliminated. Each sextant must be investigated specially for the determination of the eccentricity errors affecting the readings for different parts of the scale. These may be found by measuring a series of known angles of different magnitudes. The mean result for each angle, A, gives by (71) an equation of. the form A=R + f+E. E, (127) R two objects whose is the sextant reading for coincidence of the E and and index is the distance E, the eccentricity correction; A; I, angular corrections for those graduations of the scale which coincide with vernier and R, respectively. The readings of the graduations for the readings where R coinciding graduations when R ' the vernier reads E and is R a and R may the correction which be denoted by must be applied to R', respectively. the sextant reading, freed from index correction, in order to obtain the true value of the angle. Denoting its value by e, (127) may be written e=A Having determined e (R + f). (128) from (128) for a considerable number of angles dis- tributed as uniformly as possible over the scale, the results may be plotted as From the plot a ordinates with the corresponding values of R' as abscissas. table may be constructed giving the values of s for equidistant values of R', from which the value of s for any other reading, R, can then be derived. Care should be taken always to enter the table with the R' corresponding to the given R as argument. It should be noted that the usefulness of the table depends upon / remaining sensibly constant, for if the index correction changes by any considerable amount, Ra may change sufficiently to render the tabular values of e no longer applicable. ' The chief difficulty in investigating the eccentricity of a sextant consists securing a suitable series of known angles. simple method is to measure with a good theodolite the angles between a series of distant objects, nearly in A MEASUREMENT OF ALTITUDES 91 being taken to tilt the instrument so that in turning from one object to the next no rotation about the horizontal axis is necessary. in the horizon, care Precepts for the use of the sextant. The following points should Focus the telescope accurately. carefully be noted in using the sextant: The image of a star should be a sharply defined point; that of the sun must show the limb clearly defined and free from all blurring. For solar observations, use, whenever possible, shade glasses attached to the eyepiece rather than those in front of the mirrors; and reduce the intensity of the images as much as is consistent with clear definition. If the use of the mirror shade glasses cannot be avoided, select those which will make the direct and reflected images of the same color, and reverse them through 180 at the middle of the observing program to eliminate the effect of any non-parallelism of their surfaces. If a roof is used to protect the surface of the mercury from wind, it also should be reversed at the middle of the program. In all cases make the direct and reflected images of the same intensity by regulating the distance of the telescope from the frame. Make the adjustments in the order in which they are given above, and always test them before beginning observations. The index correction should be determined both before and after each series of settings. Make all coincidences and contacts in the center of the field. Finally, the instrument should be handled with great care, for a slight shock may disturb 'the adjustment of the mirrors and change the value 54. of the index correction. 55. The measurement of altitudes. Although the sextant may be used for the measurement of angles lying in any plane, it finds its widest application in practical astronomy in the determination of the altitude of a celestial body. At body made by bringing the reflected image of the image of the distant horizon seen directly through sea the observations are into contact with the the unsilvered portion of the horizon glass. To obtain the true reading the plane of the arc must be vertical. Practically, the matter is accomplished by rotating the instrument back and forth slightly about the axis of the telescope, which causes the reflected image to oscillate along a circular arc in the The index is to be set so that the arc is tangent to the image of the The corresponding reading corrected for index correction, dip of horizon, and refraction is the required altitude. The correction for dip is field. horizon. necessary, since, owing to the elevation of the observer, the visible horizon lies below the astronomical horizon. The square root of the altitude of the observer above the level of the sea, expressed in feet, will be the numerical value of the correction in minutes of arc. The observations are not susceptible of high precision, and the correction for eccentricity may be disregarded as relatively unimportant. For observations on land the artificial horizon must be used. The measurement of the angular distance between the object and its mercury image gives the value of the double altitude of the object. Some practice is required in order to be able to bring the object and its mercury image into PRACTICAL ASTRONOMY 92 In case the object is a star, care must be coincidence quickly and accurately. reflecare really those of the object and its taken that the images coinciding method of is the The procedure: simplest tion in the mercury. following that the mercury image is clearly visible in the the telescope toward the object. direct By bringand center of the horizon, The in field. the will reflected the zero appear image ing the index near Stand a position such in then turned slowly downward toward the mercury, the index the same time at a rate such that the being moved forward along the arc at remains of the constantly in the field. If the plane of reflected image object observer is careful to stand so that the if the and the sextant is kept vertical, its can be reflection seen, image seen directly through the unsilvered telescope is mercury will come into the field when the telescope has portion of the horizon glass Both lowered. been sufficiently images should then be visible. The varying The to change their relative positions. cause them will of the altitude object images are approaching and clamped. When they The instant of is noted and the vernier read. a the instrument determined is best coincidence slight oscillatory by giving motion about the axis of the telescope and noting the time when the reflected image in its motion back and forth across the field passes through the direct index is set so that the become coincident the time image. To obtain an accurate value of the altitude, a series of such settings in quick succession, the time and the vernier reading being should be taken It is not necessary to use the method described above for the images into the field for any of the settings but the first; for if, bringing after reading, the index be left clamped and the telescope be directed toward noted for each. the mercury image, the plane of the arc being held vertical, the reflected image will also be in the field. If it is not at once seen, a slight rotation about the axis of the telescope will bring it into view, unless too long an interval has elapsed. Measures for altitude may also be made by setting the zero of the vernier on one of the scale divisions so that the images are near each other accurately and approaching a coincidence. The time of coincidence and the vernier reading are noted. The index is then moved 20' so that the images will again be approaching coincidence. The time and the reading are noted as before and the process is repeated until a sufficient number of measures has been secured. The consistency of the measures should always be tested, as in the case of the engineer's transit (see page 79) by calculating the rate of change of the readings per minute of time. If however, the observations have been made by noting the times of coincidence for equidistant readings of the vernier, the constancy of the time intervals between the successive settings will be a sufficient test. If R denote the mean of the object will be given of the vernier readings, the apparent double altitude by (129) MEA S UREMENT OF AL TITUDES in which /is the index correction, and true altitude corresponding to the the correction for eccentricity. The of the observed times is found from e mean may be derived from Table accurate results are required, by equation (3), page we calculate z' from where the refraction, is r, 93 18. I, page If the 20, or if more zenith distance desired instead of the altitude, ' = 90 h\ (130) and z from z z' + r. (131) For measures on the sun coincidences are not observed, but, instead, the when the images are externally tangent. To eliminate the influence of semidiameter, the same number of contacts should be observed for both images approaching and images receding. If for any reason this cannot be done, a correction for semidiameter must be applied. Let instants = number of settings for images approaching, = number of settings for images receding, n = total number of settings, 5= the semidiameter of the sun calculated by equation a nr We then have for solar observations shall /,' in which (126). jp_i_ = Rd h' is wa nf c L ^~ S+I+e i / , ' f Upper sign, JLowersign, altitude decreasing. altitude increasing } "1 the apparent altitude of the sun's center corresponding to the mean of the observed times; and the term involving S, the correction for semidiameter. The true altitude and zenith distance are then given by A 2 The = A' r+p, = + r /, 2' (133) (134) may be obtained from columns four and eight of Table For results r approximate / may be taken from the fifth page and tenth columns of this same table. I, solar parallax,/, 20. Example the sun were 85. made the timepiece was tion of latitude. On 1909, April 10, the following sextant observations of the altitude of at the Laws Observatory near the time of meridian transit. The error of J0 F =+6 ra 37'. The observations will be reduced later for the determina- 94 PRACTICAL ASTRONOMT Readings on Sun CHAPTER V THE DETERMINATION OF LATITUDE Methods. 56. known means On page 34 it was shown that if the zenith distance or known right ascension and declination be measured at a time, the latitude of the place of observation can be determined by altitude of a star of of equation (31). The preceding chapter indicates the methods that be employed for the measurement of the zenith distance. It is the purpose of the present chapter to determine the most advantageous method of using the fundamental equation and to develop the formulas necessary for the practical solution of the problem. may To establish a criterion for the use of equation (31), it is to be noted that <p depend upon the errors affecting Star positions are so accurately known, however, that the <?, 2, 6, and errors in a and 8 are insignificant as compared with those occurring in z and the resultant error of observation in will . d; and we need concern ourselves only with those affecting the latter two quantities. particularly important to know the influence of an error in the time, for since this quantity is assumed to be known, it is desirable to be able to specify liow accurately it must be given in order to obtain a definite It is degree of precision in the latitude. The relation connecting small variations in z and d with changes in if is found by differentiating (31), z, t=0 and ip being considered variable. We thus find (Num. Comp. p. u.) , s\nzdz = sin dcosip dip cos 8 sin <p cos tdip cos d cos ip sin tdt, (135) i which by means of (32) and dz Writing dt=dO = cos A and solving dip (33) reduces to dip -f sin A cos <p dt. for dip = sec Adz tan A cos ipdd. (136) Assuming now that the differentials of z and represent the errors in these quantities, the resultant error in if will be given by (136). In order that this may be a minimum, sec A and tan A must have their minimum absolute when A is or 180. Since these quantities increase azimuth deviates from o or 180, the object observed for the determination of latitude should be as near the meridian as possible. Even with this limitation there will be considerable variety in the procedure depending upon the position of the star and the circumstances of the observations; and we now proceed to the consideration of the following five cases in which the values, which will occur as the given data are, respectively, 95 PRACTICAL ASTRONOMY 96 zenith distance of an object when on the meridian, difference of the meridian zenith distances of two stars, series of zenith distances when the object is near the meridian, 2. The The 3. A 1. The The 4. 5. zenith distance of an object at any hour angle, altitude of Polaris at any hour angle. I. MERIDIAN ZENITH DISTANCE on the meridian 57. Theory. The hour angle of an object this case equation (31) reduces to cos za = cos is For (137) d), (if zero. whence za or <p it is = dev (138) be derived geometrically by means of Fig. 4, p. 24 seen that the upper sign must be used for objects south of the Equation (138) whence d, <f> may also and the lower, for objects between the zenith and the lower culmination the fundamental relation becomes zenith; ^ = 180 d pole. (139) s,. For the instant of observation we have by (35) 58. Procedure. If Ad be the error of the timepiece, the clock time of transit will be O' where a, along with = a. (140) Jd, to be interpolated from the Ephemeris for the instant true zenith distance is then to be determined by some 8, is The of observation. =a For one of the methods of Section 45 or Equation (138) 55 for the clock time 6'. or (139) will then give the required value of the latitude. If a mean timepiece is used, the sidereal time of transit must be converted mean time, T, by equations (62) and (41), The clock time of obervation is then given by into the corresponding respectively. T'=TAT. pp. 49 and 39, (141) In case the error of the timepiece is uncertain, the observer will bring the image to the intersection of the threads, or the direct and reflected images into coincidence if the sextant is used, a little before the time of transit and follow with the slow motion until it becomes necessary to reverse the direction in which the tangent screw is turned in order to keep the image on the thread. This instant marks the time of meridian passage. The corresponding reading, properly corrected, then gives the altitude as before. Example 36. On 1909, April 10, an observation was made at the Laws Observatory with a sextant for the determination of the latitude by a meridian altitude of the sun. The reading on the upper limb at the calculated time of transit was 118 29' 10". The error of the clock, METHOD TALCOTT'S 97 the index correction, and the semidiameter to be used are those of Ex. 35. The calculation of the clock time of transit is in the left hand column. The reduction of the observation for the determination of the latitude is in the Gr. A. T. of Col. A. N. Sun's a at Col. A. N. J0 r second column. = 6h 9"= 14 = +6 i 0'=i 8 18' = 6*155 =118 ft 42 / 37 e i ih' 5 //' The true value of the latitude known to be 38 56' 52" i 30 /= 5 = * = 3' 8 =+7 ^ = 38 45 24 15 30 56 '5 54 29 56 19 DIFFERENCE OF MERIDIAN ZENITH DISTANCES 2. TALCOTT'S 59. == z' 1 29' 10" = -f = Unknown = 118 29 ii = 59 14 36 Theory. From equation <f (f METHOD (138) we have = $s + = #N 2s , *K. where the subscripts indicate the position of the stars with respect to the zenith. One-half the sum of these two equations gives <P=# (*. + *) + y*(zi *') + # (r t rN ), ' (142) + + which the true zenith distances have been replaced by ZK rt rK and z* The difference declinations are the and the given by respectively. Ephemeris, of the refractions is readily calculated. If therefore the difference between the apparent zenith distances of two stars be measured, the latitude can be in , calculated by (142). By limiting the application of the equation to those cases in which the zenith distances are nearly equal, a considerable increase in precision will be obtained as compared with that resulting from meridian zenith distances. Since the measures are differential, instrumental errors affecting the two observations equally will be eliminated. In the case of measures with the for the index correction and the eccentricity will be elimsextant, example, inated and need not, therefore, be determined. But what is of more importance, so far as precision is concerned, is the fact that the errors of observation which would affect these instrumental corrections, were they determined, do not enter into the result. similar condition exists in the case of the refrac- A two refractions corresponding to nearly equal zenith distances can be calculated with a higher degree of precision than is possible tion, for the difference of in the determination of the total refraction. methods of uated 7 circle. Finally, the fact that the quantity small, makes it possible to introduce other and more precise measurement than those which depend upon the use of a grad- to be observed is For example, with the engineer's transit small differences of PRACTICAL ASTRONOMr zenith distance may be measured more accurately with the gradienter screw than with the vertical circle. The method under discussion was first proposed by Horrebow, the director about the middle of the eighteenth century, Copenhagen and was given extensive practical application in the work of the United States Coast and Geodetic Survey about a century later by Captain Talcott, from which circumstance it is commonly known as Talcott's method. It reaches its highest precision when used in connection with the zenith telescope, an instrument of the altazimuth type fitted with an accurately constructed micrometer eyepiece and a very sensitive altitude level. The level enables the observer to give the line of sight the same inclination to the vertical during both observations, while the micrometer affords a very precise determination of the Observatory of of the required difference in zenith distance of the two stars. If the method is to be used in connection with the engineer's transit, the angular value of one revolution of the gradienter screw should first be determined by measuring a small angle whose value is known. The observa- made and reduced in a way such that any irregularity in the To this end a process analogous to that used in be 26 and 29 may employed. Examples Since the correction for refraction will always be small, we may assume tions should be screw From will (4) be revealed. we find dr j~ 00 = 57 , . sec'.s sin o i , which expresses the rate of change of r per i of change in z'. Denoting quantity by C, the correction for refraction in seconds of arc becomes %( r ')"=&(%' z')C this (143) which the difference of the zenith distances must be expressed in degrees. value of C may be taken from Table V with the mean zenith distance of the two stars as argument. in The TABLE *' V CIRCUMMERIDIAN ALTITUDES 99 and calculate the clock time of meridian transit by (140) or (141). If a sextant is used, measure the double altitudes of the two stars at the Let Rs and RM be the corresponding sextant readings. instants of transit. The second term of (142) will then be given #(*.' by *,,)=#(*..-*.) (145) If the engineer's transit is employed, level carefully and bring the star culminating first to the intersection of'the threads at the instant of its transit. Read the gradienter screw, reverse, relevel, bring the second star to the intersection of the threads at the instant of transit by means of the screw, and note the reading as before. The vertical circle should be firmly clamped when the setting on the first star is made, and must not be disturbed thereafter If the two screw readings be until the second star has been observed. denoted by m s and m a and if G be the value of one-half a revolution of the screw, we shall have , Yi(z'. which the upper sign in is to s') = /), G(m, (146) be used when the screw readings increase with increasing zenith distance. In levelling, special attention should be given to the altitude level. Unless the bubble has the same position for both observations, an error will be introduced into the result. If the level is a sensitive one, it will be better to omit the levelling after reversal and apply a correction to the result given and be the readings of the object and eye ends of the if readings increasing toward the north be recorded as positive while those increasing toward the south are entered as negative, the correction to be added algebraically to the result given by (146) will be by If (146). o e bubble, respectively, and (o.+e.+o.+ e n )D, (147) D is one-fourth the which angular value of one division of the level. The bubble readings should be taken as near the times of transit as possible. The last term of (142) is given by (143), the value of C being derived from Table V. The declinations are to be taken from the list of apparent places in in the Ephemeris for the instant of observation. its declination served at lower culmination, <J S 180 In case the northern star in (142) is ob- must be replaced by . 3. CIKCUMMERIDIAN ALTITUDES The zenith distance to be used in equations (138) and (139) when on the meridian. Since only a single determination of this quantity can be made at any given transit, it is desirable for the sake of precision to modify the method described tinder No. I so as to permit a multi61. is Theory. that of the object plication of the settings. The change in the zenith distance during an interval immediately preceding or following the instant of transit is small and its value is easily and PRACTICAL ASTRONOMY 100 The meridian accurately calculated. by observing when the object zenith distance therefore be found may near the meridian and applying to the measured value of the coordinate the amount of the change during the interval series of separating the instant of observation from that of culmination. such measures reduced to the meridian gives a precise value of z which can then be substituted into (138) or (139) for the determination of the latitude. is A It is of course immaterial whether the quantity measured be zenith distance or The method is commonly knqwn as that of circummeridian altitudes. altitude. The development, of the formulae reduction to the meridian is to be used for the calculation of the as follows: Equation may (31) be written in the form = cos cos 2 cos 8) ((f> y cos d sin' ^ /. (148) Let z be the observed value of the coordinate, # the meridian value, and We then have Zthe reduction to the meridian. z+Z=z (149) a. Substituting into (148) we find Z) =cos sa cos (za To express Z explicitly we may A = cos and neglecting terms in <p Z 3 we have to the t. (150) m = 2sin'%t, , (151) find Z= Squaring, Yz Z cos d cosec z we cos 8 sin* ip replace the left member of (150) by its expanis small the convergence will be rapid. by Taylor's theorem. Since Introducing at the same time sion 2 cos Am+^Z'cots,. (152) same degree of approximation Z 3 = A'm*. Substituting into (152), and writing B=A*cots we have finally for the Q, n=y nf=2s\n*y 2 t t, (153) reduction to the meridian. Z= Am+n. Since the observations may be arranged so that the error in (154) will be insensible. Combining equations becomes (138), (149), <p and (154) Z (154), the = 8 z^Am n, will not exceed 15' or 20', expression for the latitude 055) C1RCUMMERIDIAN ALTITUDES 101 which the upper sign is to be used for southern stars; and the lower, for those culminating between the zenith and the pole. For an object observed near lower culmination, / in (31) must be replaced /. The resulting value of the reduction to the meridian substituted i8o in + by into (139) gives <p = 180 3 z Am Bn. (156) Equations (155) and (156), in which the last terms are to be calculated by For observations with (151) and (153), express the solution of the problem. the engineer's transit the term Bn will usually be insensible when the hour m than 15'" or 20 the quantity It will be observed that A and B depend upon the latitude value of if sufficiently accurate for the calculation of to be determined. these coefficients may be obtained by (138) or (139) from the value of z angle is less . A observed nearest the time of transit. It will be noted further that A and B are constant for any given series of observations and need be calculated but once. on the other hand, are different for each setting. Since and The factors they depend only upon the hour angle, their values may be tabulated with t as argument. Tables VI and VII may be used for all ordinary observations with m , the transit or sextant. TABLE VI t TABLE VII PRACTICAL ASTRONOMY 102 zenith distance observed nearest the time of transit for za and for the determination of an approximate value of <p both of which are required for the commeans of Finally, calculate the latitude for each observation by putation. used is that corresponding to the instant to be The declination or (156). (155) of observation. The final result may also be obtained by applying the mean of all the , Am and of Bn to the mean of all the zenith distances in accordance with equations (155) or (156). This method, however, gives no indication as to the consistency of the observations, and it is better to reduce the results the means of not more than two separately, or, at least, to reduce separately or three consecutive measures. values of The method of circummeridian altitudes may advantageously be combined with that of Talcott. When this is done there will be given a series of derived from observations made near the meridian. values of % (z' s Each of these must be reduced to the meridian by adding to (142) the term O y2 (Z ZN s ), in which Z s and ZN are to be calculated by (154). Example 37. The reduction of the circummeridian altitudes given To eliminate the semidiameter the means are formed for the follows: in Ex. 35, p. 93, is as ist and 2nd, 3rd and These results are in the first and sixth 4th, 5th and 6th, and the yth and Sth observations. The eccentricity corrections are unknown. The index corlines of the calculation below. rection found in Ex. 35 jh fjm js. / (sidereal) t (solar) m is -j- i". In Ex. 36, p. 96, the clock time of transit was found to be LATITUDE FROM ZENITH DISTANCE 103 This formulae for circummeridian altitudes no longer give convergent results. is readily accomplished by using the fundamental equation (31) in the form cos z = cos A7 (<p (157) ). Equation (157) is the last of equations (34), the auxiliaries defined by the first and second of this group. 64. Procedure. Having determined object, calculate the hour angle by the true zenith distance of the t=0-a, in which is n sin and tp (158) the true sidereal time of observation. n cos N= sin and A7 being Then determine n and A7 by d, A7 = cos d cos fica\ t, N from cos 2 (160) A A7) must have the reference to the fourth of (34) shows that sin (^ A7) from same algebraic sign as cos A. This together with the sign of cos(^ 7 A The latitude is then given by (159) determines the quadrant of <p . <p = (tp N) + N. (161) Equations (i58)-(i6i) are rigorous and apply to all values of the hour angle, but care should be taken to observe as near the meridian as possible in order that errors in z and d may not appear multiplied in the result. (See Section 56.) A sufficient number of decimal places must be employed to offset A7 is determined from its cosine. the fact that the angle <p Example 38. On 1908, Oct. 2, at watch time 8 h 35 m n P.M. the altitude of Polaris was found to be 39 2g'.8. (See Ex. 31, p. 79.) The error of the watch on C.S.T. was -)- i m 45*. Find the latitude by equations (158) -(161). 8 h 36 m 56" C.S.T. Columbia cos,) cos* 21 13 14 a i 27 10 t 19 46 4 wsinA 296 3i'.o tan A 50 30.2 A 5 3i-3 lo g 88 49.0 cosz t z' calculated ' " i.i sin A^ cos(y> The " 7 7 7 r ij cos AT ^ is larger than the true value by o'.6 <p if A^) A^ 8.3150 9-6498 7.9648 9-9999 a. 0351 89 a8'.3 o.oooo 9-9999 9-8033 9.8034 50 30'.$ 38 57.5 Ant. PRACTICAL ASTRONOMY 104 The C.S.T. is converted into the corresponding Columbia by (41) and (58). a and J are from p. 321 from the Ephemeris. The value of t shows that Polaris was east of the merand sin (y> idian at the time of the observation, whence cos IV) are negative. Since cos N} is positive, y> ^V is in the fourth quadrant. A (tf ALTITUDE OF POLARIS 5. The peculiar location of Polaris with respect to the pole possible to simplify the fundamental latitude equation for use in connection with this object. Since the latitude is by definition equal to the 65. makes Theory. it altitude of the north celestial pole, the problem may be solved by finding an in altitude of the pole and Polaris. The polar expression for the difference distance of Polaris is about i To always be a small angle. n', consequently, the required difference will is due the possibility of a simplification this fact of equation (31). (See Num. Comp. pp. 14 and 16.) Replacing z and d in (31) by the altitude, ft, and the north polar distance, IT, respectively, we find sin If ffbe the difference h = cos n sin + sin <p in altitude of Polaris <p=h tpH Writing h in (162), sin Since at a we may H= TT cos ip cos /. and the pole, we shall + H. -\- tan have (163) and expanding and solving for sins- cos t (162) ^(cos// sin cos/r). H (164) maximum replace sin not exceeding o"3. Hand At cos with errors which are = still by //and TT, respectively, with an error same time we may write sins- in (164) the ! smaller, thus obtaining H= -ncost + yztanyfr* Neglecting terms involving n3 H'). (165) , H' = Tf cos 3 /, and substituting H' into (165) we have H= Finally, by (163) -7z-cos/ + ^7r a 2 tan^sin /. (166) LATITUDE FROM POLARIS <p in =h K cos / + 105 (167) A!", which " K%x' tan tp sin*/. (168) error in the latitude calculated from (167) due to the approximate of the form equation will usually be less than 2" the quantity to be The calculation of requires a knowledge of <p coefficient is about since the 0.02, a rough approxdetermined; but, J^/r* only The K K imation for the latitude will answer. The values of may be derived from Table VIII with an approximate latitude and the hour angle as arguments. i 1 i' o". The table is based on the value it TABLE VIII t K= tan a sin* t PRACTICAL ASTRONOMT 106 Find the latitude by equations (169) from the data of Ex. Example 39. TV 88 S 10" 9" 42 jrw * -32 49'. i h' 70.-9 * i- TTCOS; 57.0 C. S. T. 9 41 38 logr 1.8506 Columbia 23 i -9 cos 9-95 6 a i 27 * 21 34 t ' log ^ cosi 13 16 39 ^ 1.7562 The 323 34- calcu i ate d 9 is 54'. 9 - 3s <f The hour angle o'. I. = ^ 38S7'.S. is is to be calculated as before. to Its value in both cases be interpolated from Table IV, Ephemeris, Consequently, argument. We then find Ex. 38 24 h / 4 h H <f is h greater than I2 p. 595, -31-2 38 57-S 38 56.8 We produced by a small change d(f all of the preceding and we may write in is = by tanAcosydO. A / . as more closely The change (136) A (170) be small, a few degrees (170) with sufficient Substituting for sin A its value from (33), methods but No. sin with 24" 2577 3953'-S 57-0 may now examine 67. Influence of an error in time. the influence of an error in the time upon the calculated latitude. For which agrees Ex.39 2h i39 3928'.7 // at most, ^*- Find the latitude by means of Table IV of the Ephemeris from the data Example 40. (p 2 57- of Exs. 38 and 39. is 1 larger than the true value by The application of equations (169) to the data of Ex. 38 gives the formula of Section 64. exactly with the result obtained by H 33, p. 81. in 4, will place of tan accuracy for the present purpose. and writing z equal to the meridian zenith distance, A in , (170) becomes sin tdO. in which the upper sign refers to southern (171) reduces by stars. For circummeridian altitudes (151) to d<p For Polaris we have with = sufficient = 90 A sin t dO. (172) approximation <p, COS <? = = 0.02, 7T whence d<p= 0.02 sin tdO. (173) Equations (172) and (173) may be obtained directly from (153) and (157) by differentiating with respect to t and introducing rf/ dO, the small terms = Bn and' AT being disregarded. INFLUENCE OF ERROR IN TIME 107 Equations (170) -(173) may be used for the calculation of dtp when dO is known, or for the determination of the accuracy with which the time must be known in order to obtain <p with a given degree of precision. If dO is expressed in seconds of time, the factor 15 must be introduced into the right members of the various equations in order that dtp may be expressed in seconds of arc. It is evident that, aside from the dependence of dtp upon t, it also depends upon the zenith distance and declination of the star, and that an error in the time has the least influence upon the calculated latitude for stars near the For Polaris the effect of dO is always small, and if t be near oh or I2 h it pole. , be very slight indeed, even though dO be large. This fact taken in connection with the simplicity of the reductions renders the last of the above methods the most useful of all the various processes that will be employed for the determination of latitude. The greatest precision, however, is attained only by the method of a Talcott when used in connection with the zenith telescope. may Example 41. What from the first of the circump 101, on the assumption that the watch correction used was the error in the latitude calculated is meridian altitudes of Ex. 37, incorrect by 20"? By (172) we find, taking the values of do Example 42. Polaris given than o'. How A and = Kf = 300" / t from Ex. 37 = 7>"39 = i log^l sin/ 0.1744' log dy 1.1849 8.5334 <fys='S" log 300 2.4771 54'45" accurately must the time be known in order that the altitude of may yield a value of the latitude uncertain by not more In Ex. 33, p. 81, i? By 073) ar) d the data in Ex. 39, p. 105, ' sine 3 23 34' 0.595 rf0 we find <fy o. 02 sin/ = 8'.33=33" Ans. o'.i 0.012 CHAPTER VI THE DETERMINATION OF AZIMUTH The azimuth Methods. 68. of a terrestrial mark may be found by ob- the mark and a celestial object and applying serving the difference in azimuth of to the into this difference the calculated azimuth of the object corresponding The methods to be employed for the observational part stant of observation. We of the process have been discussed in detail in Chapter IV. examine the means by which the azimuth of the celestial body puted. A have may now to be com- _ _ to the fundamental rigorous and general method of procedure leading equation cos d sin sin d cos tp t cos d sin if cos t Before proceeding to the adaptation of this equation 34. under purposes of calculation it is desirable to investigate the conditions which it may most advantageously be employed. The calculated azimuth will was outlined on page to the depend upon the right ascension and declination of the star, the time, and the quantities may be assumed to be known with precision, but the last are likely to be affected by relatively large To determine the influence of these upon the calculated azimuth, uncertainties. latitude of the place of observation. The first two and thus derive a precept for the choice of objects to be observed, we differentiate (33), A, z, and t being considered variable, and substitute for dz its value dd we find after simplification. from page 95. Writing at the same time dt = dA = cot 2 sin A dip + (sin z sin y + cos z cos tp cos A) cosec z dd. Fig. 6 we denote the angle at by q, the expression in parenthesis reduces by the second of the fundamental formulae of spherical trigonometry to cos d cos q, whence If in dA In order that cot z sin dA may A dp + cos d cos q cosec zdO. 075) it is necessary that the object should not the factors cots and cosec s will produce a multiOtherwise, and dd. is it desirable that the azimuth should Further, dip be small be near the zenith. plication of both be near o or 180, for when this is the case an error in the assumed latitude When the object is upon near the pole, cos d will be small and the influence of dd will be slight; and if, at the same time, it be near cos will also be small, and the effect elongation, q of dd will still further be minimized. will produce but A little effect the calculated azimuth. close circnmpolar star at any hour angle satisfies these conditions with render the influence of any ordinary errors in ip and Should the clock correction be very uncertain, however, it may sufficient closeness to quite insensible. 108 AZIMUTH OF THE SUN 109 be desirable to observe for the determination of the azimuth difference of the mark and the star at or near the time of elongation in order that the coefficient of dd in (175), already small through the presence of cos 3, may be made still smaller by the introduction of a value of q near 90. Far less satisfactory will be the result in the case of observations on the may sun, although this object be used when the latitude is known with some to observe as far from the meridian as posprecision, provided care be taken in (175) depending on s and q will coefficients the this With sible. precaution minimization of the errors in <p and d, especially for a values best the have adapted that of the latter, which in all cases is most to be feared. Besides the fundamental equation (174) there is another which times useful, namely, If (26). the distance of the zenith celestial is some- body be measured simultaneously with the determination of the azimuth difference, the azimuth of the body may be calculated by this equation, whence the azimuth of the mark can be found as before. With this method of procedure the latitude of the place must be known, but the time does not enter into the problem except as it may be required for the interpolation of the declination of the object for the instant of observation. To determine the conditions under which this method advantage, differentiate find after simplification cos considering (26) if dA s, <p, = cos q cosec / dz and A may be used with as variables. We thus ('76) cottd<p. From and <p will have the least influence this it appears that errors in and q are as near 90 or 270 as possible. These conditions cannot both be fulfilled at the same time. But for circumpolar stars observed near elongation the magnitude of cosq and cotf in (176) will be such that errors in z and <p will have only an insignificant influence on the calculated azimuth. when t consideration of the preceding results indicates that we shall need adaptations of the fundamental azimuth equations designed for the calculation of The 2. The azimuth The azimuth 3. Azimuth from an observed zenith 1. of the sun. of a circumpolar star at any hour angle. I. distance. AZIMUTH OF THE SUN 69. Theory. The first four equations of (34) are the equivalent of (32) and (33) from which the fundamental equation (174) was derived. By their combination we find the following group which for the purposes of calculation replaces (174)- tan = -cos (J tan AT (177) -, / r . sin The quadrants of TV and algebraic signs as sin /J and A sin .... (tp are determined t, respectively. tan/. (178) N) by noting that sin JV and sin A have the same PRACTICAL ASTRONOMY HO Procedure. 70. in which is If a sidereal timepiece is used, calculate and the true sidereal time of observation, from t the sun's apparent If a mean solar timepiece is employed, calculate the apparent right ascension. This is disolar time for which the azimuth difference has been measured. a when and the sun. of d, required for the the hour Interpolate angle rectly for the instant of observation. Finally compute A from (177) from solar observations should be made determinations Azimuth and (178). is far from the meridian. the sun when only calculation of 2. t, AZIMUTH OF A CIRCUMPOLAR STAR AT ANY HOUR ANGLE <f and Dividing the numerator IT, we find 90 Theory 71. sin d cos denominator of (174) by and writing d tan^ tans- sec =i sin/ <p tan n tan cos <p (179) , t This equation may be replaced by the following group which reference to the requirements of calculation. g = tan TT sec 7i arranged with <p, = tan tan = gs\n c^ h cos tan A = gG sin h is ip <f, i (180) /' i t. The quadrant sign as sin of A is determined by the fact that sin A must have the same algebraic t. The factors g and h are constant for any given night, and in approximate work they may be considered as such for a series of nights. Moreover h is small because of the factor tan;r. G therefore differs but little from unity, and the values of logG may be tabulated with log h cos t as argument. Such a table, sufficient for all practical requirements is given in Rept. Supt. U. S. Coast and Geodetic Survey, 1897-8, pp. 399-407. In case tables for follows as cost G : is logG are not accessible has the form i/ (i +v) negative or positive. The or i/ latter (i its values v), in expression may which v may be calculated as = h cos be written G=i/(iv) = (i+v)(i+V)(i+V). ... t. according in the form (181) Since v is small, the parentheses after the second or third in the last member of (181) will sensibly be equal to To find the value of logG, therefore, unity. we must find the logarithms of one or more factors of the form (i+fc). For this purpose we use the addition logarithmic table. Since i, the formula a= are (Num. Comp. p. 10). AZIMUTH OF A CIRCUMPOLAR STAR A = ldgb, where B is to be interpolated 111 log(l+&)=B, from the table with A Hence as argument. For cost negative, A For cost A, log (h cost), logG = B. (182) positive, = log (A cos /), A, = log (k cos /)", A 3 = log (A cos /) ---- Equations (180) used in connection with tables for logC, or with formulae afford a convenient and precise method of calculating the azimuth of (182) any of the close circumpolar whose apparent places are given stars in the Ephemeris, pp. 312-323. Equations (180) are rigorous, however, and for approximate results they may be simplified, especially if the circumpolar observed is Polaris. For this n', and for latitudes less than 60, its azimuth object JT at the present time is i We may therefore write will always differ from 180 by less than 2 3'. 180 with an error not exceeding 2". be less than A For =jrGsec^sin latitudes of i". TABLE IX t 45 t (183) or less the error will always PRACTICAL ASTRONOMY 112 between the value of tp for the calculation of the table assumed and that cor- have only a slight influence on the responding to the place of observation will be understood, however, that the local azimuth derived from (183). It is to of the coefficient n appearing value the and that value of sec^ must be used, In case a number of of observation. (183) must correspond to the date the at a made are to be corresponding local azimuth determinations given station, mean values of with the combined be sec value of log <p may conveniently from the table. sec G then One can f directly interpolate log log G. i 10'. 40, and The values of log G in Table IX are based upon <p maximum The for north distance mean is which the the latter of 1910.0. polar in = from the u?e of absolute errors in the azimuth resulting n= this table for various latitudes are Latitude Error in The values of log tude of the the G sec f 30 35 o'.24 o'. 12 in the third Laws Observatory, which Procedure. 72. A is 38 45 o'. 50 15 o'.38 column of Table IX refer to the lati- 57'. and d Interpolate 40 o'.oo for the instant of observation of apparent places of circumpolar stars, Ephcweris, pp. 312-323. list from Cal- culate the true sidereal time of observation for which the azimuth difference and the mark has been measured. Then For a precise azimuth, calculate A from (180). The value of logG may be taken from Kept. Supt. U. S. Coast and Geodetic Survey, 1897-8, pp. 399-407, or from some similar table, with the argument log h cost; or it may be calculated by means of (182). For an approximate azimuth from Polaris, interpolate log G, or log G sec <f>, from table IX with t as argument. Then calculate A from where d is of the star : A= If given JT in TrGsecysint. be expressed in minutes of minutes of arc. Example 43. The i8o arc, the last term of (184) Determine the azimuth of the mark from the data given latitude of the place of observation is 38 (184) will also be in Ex. 34, p. 85. 56' 52". Equations (180) are used for the calculation, the results for the two positions of the instrument being reduced separately. The azimuth of the mark is found by subtracting the difference 5 M, taken from p. 85, from the calculated azimuth of Polaris. The difference of the two values of is not to be taken as an indication of the precision of the result, as these quantities are affected by instrumental errors whose influence is not eliminated until the mean is formed. M AZIMUTH FROM ZENITH DISTANCE i i if 3 h 25 19" 10' 46" 56 5i sec/p 0.10918 tan?r 8.31362 log// 9.90756 8.42280 8.22118 113 PRACTICAL ASTRONOMY 114 zenith distance determined simultaneously with the measurement of the azimuth difference, the declination for the instant of observation, and the latitude of the place of observation constitute the data necessary for the calculation of the The azimuth. For objects whose azimuths are not so near o or 180 as to render the error But for of calculation for (185) large, we may calculate A by this equation. use with the method in question, it stars, which are best adapted for circumpolar In any case, however, (185) will be desirable to derive A by means of (186). and (186) will serve as a mutual control for testing the accuracy of the calculated azimuth. M, the azimuth differ75. Azimuth of a mark.- Having measured 5 the azimuth of the determined and having ence of the object and the mark, we calculate methods M, the azimuth of the object by some one of the above mark, by M=A (SM\ (i86a) Example 44. Find the azimuth of the mark from the data given in Ex. 33, p. 81. Since both the time and the altitude of Polaris corresponding to the instant of measurement of the azimuth difference of the star and the mark are known, the reduction may be made by the third as well as by the second method. The first column contains the calculaThe value of t required for the first part is taken tion by (184); the second, that by (186). from Ex. 39, p. 106. t INFLUENCE OF ERROR IN TIME = cos q sec dA <p cosec 115 / dz. (188) Equations (187) and (188) may be used to estimate the uncertainty in A and z, or they may be used to corresponding to a given uncertainty in determine the accuracy with which the time or the zenith distance must be known in order to secure a given degree of precision in A. Usually z and tion of The dA. t may be estimated with sufficient precision for the derivamay be calculated from parallactic angle q sec (f g= sin A sec S sin sin /cosec z. ( l ^9) For circumpolar stars (187) and (188) may be simplified as follows: Since the azimuth of such an object is always a small angle, the spherical excess of the triangle PZO, Fig. 6, page 26, is small and we shall have approximately q= 1 80 t, whence cos^ = cost. (190) = go Further, we have with sufficient approximation z these results into (187) and (188) and writing cos<J = dA The first writing of these can G= also = ;r, <p. we Substituting find (191) sec^cot/dfc. (192) be derived from (184) by differentiating and i. Example 45. The altitude of the sun and the difference of its azimuth and that of a mark were measured with an engineer's transit at the Laws Observatory on 1909, April 27. m The results were 7\v 4 h I" n'o, P.M., J7\y = i = 33 i<)'.(>, 44'S (referred to C.S.T.), = 1 ' S Af=8i24'.7. Find the azimuth of the mark, calculating the azimuth of the sun both by method i and method 3. The computation of the solar azimuth by (177) and (178) is in the first column; that for In the latter instance the time is required only with such precision as (186), in the second. as may be necessary for the interpolation of declination from the Ephemeris for the instant of observation. C.S.T. 3 59" 26:5 Col. M.S.T. 3 50 8.2 r i 24.7 a 52 32.9 E t = Co\. A.S.T. / 5 tan ,J cost tan N 58 8'.2 +13 3., 9-7"55 sin(s a) cosec s 0.00025 9.66941 13 54.7 A 8.9 51 56.9 9.81604 2'.2 tan^l 76 c) 56.9 ((f>JV) = oo-y, * 25 tan/ 41.7 sin(i 38 N 56 Si-' <p cos TV I9'.6 1.3 =z = ,r c 33 p 9-39196 -W y cosec 3 /*' 9-9S7I5 0.20651 0.61902 0.78268 80 38'.! cosec 91 (. cot%A A S M M *) 9.76132 0.56498 0.07130 80 38:2 Ck. 81 24.7 359 13. 5 CHAPTER VII THE DETERMINATION OF TIME Methods. The determination of time means, practically, finding 77. the error of a timepiece. To accomplish this the true time 6 or T\s calculated from observations on a star or the sun and compared with the clock time at The required which the observations were made. J0=0 0' error is given by (193) t or jT=rr, according as the timepiece is mean sidereal or (194) solar, 6' and T' being the clock values of the time of observation. The fundamental equation for the determination of time = +7. is (195) Applied to any celestial object this equation gives the sidereal which the mean solar or apparent solar time may be derived by the ation processes of Chapter III. For the sun, however, the hour the solar time, and, in case of observations on this directly apparent mean solar time may be found from (42) written in the form T=t + E. time, from transform- angle t is object, the (196) When the timepiece is solar the use of (196) is simpler than that of (195). Since a and E may be regarded as known, the problem is reduced to the determination of the hour angle of the object for the instant of observation. As indicated on page 34 this may be accomplished by measuring the zenith distance of the object at a place of known latitude and using equation (38) or (39)- The problem can ' also be solved of the by determining the clock time the object is zero. For this case the instant for which the hour angle of fundamental equation reduces to 6 = a, (197) and J6 In outlining the ' it will sidereal. =a d a '. (198) methods that may be employed for the determination of is a star and that the timepiece used is be assumed that the object The modifications necessary be considered in for the removal of these limitations will connection with the discussion of the details presented in the following sections. 116 METHODS To determine #' we may note the time 117 6, when a star has a certain zenith distance, or altitude, east of the meridian, and, again, the time 0, when it has the same zenith distance west of the meridian. Since the celestial sphere rotates uniformly, we shall have 0: = x(o, + (199) o.). The method is known as that of equal altitudes. The clock time of meridian transit, may also be determined by noting ', the instant of passage of an object across the vertical thread of a transit instrument mounted so that the line of sight of the telescope lies in the plane of the meridian. This is the meridian method of time determination. ' Finally, may be found by observing the transit of an object across the vertical thread of an instrument nearly in the plane of the meridian. The application of a small correction to the observed time depending upon the displacement of the instrument from the meridian gives the clock time for which f o. In practice the deviation of the instrument is such that the line of sight lies in the plane of the vertical circle passing through Polaris at a definite instant. The process is accordingly known as the Polaris vertical circle method of time determination. It is of special interest on account of the fact that it is readily adapted to a simultaneous determination of time and = azimuth. There are other methods of determining the true time, but those outlined afford a sufficient variety to meet the conditions arising in therepractice. fore proceed to a detailed consideration of We The zenith distance method. The method of equal zenith distances The meridian method. The Polaris vertical circle method. 1. 2. 3. 4. or altitudes. THE ZENITH DISTANCE METHOD I. Theory. The formulae necessary for the calculation of / from S, <p, were developed in connection with the discussion of coordinate transformations and are given in (38) and (39). 78. and z, The a, 3, <p, From resultant error of observation will and (136) z. we Those in a and 8 we may depend upon the errors affecting disregard as relatively insignificant. find dd = cosec A sec <p dz cot A sec <p d<p. (200) that dz and dtp represent the errors in z and and dO the resultant tp, error of observation in 0, it that for a latitude the time will be appears given least affected by uncertainties in z and when the azimuth of the object is <p near 90 or 270. Care should be taken, therefore, to select for observation Assuming only those objects which are near the prime vertical. PRACTICAL ASTRONOMY 118 Having found the true zenith distance corresponding by (38) or (39). The latter equation should not Procedure. 79. to the clock time, calculate t be used when the object is so near the meridian that the interpolation of its cosine is rendered uncertain. / from Observations on a star: If the timepiece is sidereal, calculate by (195), convert the sidereal time derived from (195) into the solar, (193); corresponding mean solar time T, and determine AT from (194), taking care and JO by T is that if reduced to the meridian to which the clock time Observations on the sun: If the timepiece is sidereal, refers. we may proceed as the case of a star using (195) and (193), or we may convert the value of T derived from (196) into the corresponding sidereal time and then use (193). in If the timepiece is solar, calculate T from (196), reduce its value to the meridian to which the clock time refers, and calculate JT'from (194). Owing to the change in the right ascension and declination of the sun, a knowledge of the approximate time is necessary for the reduction of solar observations. Should the error of the timepiece be unknown, the interpolation of a and 3, or may be made with the Greenwich mean time corresponding to , the clock time of observation. The resulting data will give an approximation sufficient for a precise for the error of the clock which, in general, will be A interpolation of the coordinates of the sun. repetition of the calculation then gives the final value of the clock correction. Example 46. Find the error of the watch from the measured altitude of Alcyone given in Ex. 31, p. 79. We have h' 21 19' 30" 2 / i8 h 35"> j.6'7 a 3 22 42 3.2 17 19.9 51.4 h 21 17 30 o '5 +23 49 23 C.S.T. 9 40 38 56 52 Watch 9 39 6.4 +1 45.0 <f J7\v = The The solution of (38) gives / 18*35 l6 !8. value used for / is the mean of these. C.S.T. is Ans. From (39), as a control, we find i8 h 35 m 16-6. The conversion of into the corresponding accomplished by (62) and (41). 2. THE METHOD OF EQUAL ALTITUDES and be the sidereal clock times when a star has the and west of the meridian, respectively, the clock time of meridian transit will be given by (199), whence by (198) 80. same Theory. If 0, d, altitude, or zenith distance, east JO If a solar timepiece is = a#(0 + O I (201) m ). used we shall have JT= T- X (T, + T,), where T is the mean solar time corresponding to (202) 0=a. the object observed is the sun, the above equations are not applicable on account of the change in the declination during the interval separating the If TIME FROM EQUAL ALTITUDES 119 be included, however, by reducing the observed times to what they would have been had the declination been constant and equal to its value at the instant of meridian transit. Since the change in d is small, the required corrections may be found from the differential relation connecting changes in d with corresponding changes in t. From (31) measures. This influence may tan 3cott)dd, dfr=(tany>cosec/ (203) one-half the interval between the two observations expressed in the declination for apparent noon, and dd the change in 8 during the interval /. Both the observed times will be too late by the quantity dt. in which / is solar units, Hence, <? made for solar observations M=a If the timepiece is with a sidereal timepiece, y2 (0 + 6,) + dt. we have from solar, (204) t and (196) (202), since / = for the instant of meridian transit, AT=E#(T + TJ + l It is dt. (205) sometimes convenient to combine afternoon observations with others made on the following morning. In this case the mean of the observed times corrected for the change in declination is the clock time of lower culmination. The quantity / in (203) is one-half the interval between the observations expressed in solar units as before; but d must be interpolated from the Ephemeris for the instant of the sun's lower transit, and the resulting value of dt must be added to the clock times of observation. The expressions for the clock correction are J0=l2 h + jr=i2 + h in which the values of a and E + y (T + T,) tf (0, 2 0,) s dt, (206) dt, (207) refer to the instant of lower culmination. Procedure. The object observed should be near the prime vertical. three or four hours east of the meridian note the time of transit across the horizontal thread of the transit for a definite reading of the vertical circle, 81. When most conveniently an exact degree or half degree. Change the reading by 10' or 20' and note the time of transit as before. Repeat a number of times, always changing the reading by the same amount. After meridian passage observe the times of transit over the horizontal thread for the same readings of the vertical circle as before, but in the reverse order. If the sextant is used, note the times of contact of the direct and reflected images for the same series of equidistant readings of the vernier before and after meridian passage. Denote the means of the two series of times by 0, and d,, or t and T,, according T as the timepiece is sidereal or mean solar. For a star the error of the clock will be given by (201) or (202). For the sun, calculate dt by (203), and the clock error by (204) or (205) in case the observations are made in the morning and after- PRACTICAL ASTKONOMT 120 noon of the same day, or by (206) and (207) when they are secured in the afternoon and on the following morning. Care must be taken not to disturb the instrumental adjustments between the two sets of measures. If these remain unchanged no correction need be or semidiameter. applied for index error, eccentricity, refraction, parallax This fact taken in connection with the simplicity of the reductions constitutes It is subject, however, to the serious the chief advantage of the method. hours must elapse before the observing objection that an interval of several can be completed, with the danger that clouds may interfere with the program second series of measures. the engineer's transit is used for the observations, all the measures should -be made in the same position of the verticle circle, and the angles When should all be set from the same vernier. method of time determination, an observed is approximate knowledge of the time is necessary when the object the sun. If the clock correction is quite unknown, this may be derived from As in the case of the zenith distance only necessary to interpolate the sun's right ascension, or the equation of time, as may be required, on the assumption that the clock error is zero. This approximate result will lead to an approximation for the error of the timepiece with which the calculation the observations themselves as before. may be It is repeated for the determination of the 3. 82. Theory. a transit instrument final value. THE MERIDIAN METHOD The meridian method of time determination mounted so that, when perfectly adjusted, the requires line of sight lies constantly in the plane of the meridian, whatever the elevation In order that this may be the case, the horizontal axis of the telescope. must coincide with the intersection of the planes of the prime vertical and the horizon, and the line of sight must be perpendicular to the horizontal axis. The instant of a star's transit across the vertical thread will then be the same meridian passage. Denoting the clock time of this instant by 6 the error of the timepiece will be given by ' as that of its J0= (208) '. general, however, the conditions of perfect adjustment will not be The horizontal axis will not lie exactly in the plane of the prime In satisfied. When produced it will cut the celestial azimuth referred to the east point whose sphere page 65, and whose altitude we may denote by a and t>, respectively. Further, the line of sight will not be exactly perpendicular to the horizontal axis, but will form with it an angle 90 The quantities a, d, and c are known as the azimuth, In general, therefore, the star level, and collimation constants, respectively. vertical, nor will in a it be truly horizontal. point A, Fig. 8, +. will not be on the meridian at the instant of its transit across the vertical thread, but will have a small hour angle t whose value will depend upon the magnitude of the instrumental constants a, b, and c and the position of the star. To obtain the clock time of meridian transit clock time of observation, 6', whence we must subtract t from the THE MERIDIAN METHOD e,' = B' 121 t, (209) 0'-\-t. (210) and by (208) J6 The values of a, b, and =a Consequently J0 can be be found. c can always determined by (210) when / has been mental constants. For this purpose we make use of equations (82), (89), and The last two terms of (82) express the influence of the level and colli(33). mation constants, b and c, upon the reading of the horizontal circle of the engineer's transit for C. R., or, what amounts to the same thing, the amount by which the azimuth difference of the point A and the object 0, when on the The last two terms of (89) express the corresvertical thread, exceeds 90. ponding quantity for C. L. These results may be applied directly to the meridian transit to determine the azimuth of the star at the instant of its transit across the vertical thread. For, denoting this azimuth by A s and expressed as a function of the instru- , assuming that a, the azimuth of the point measured positive toward the south, we have A, = a + b cot z A referred to the east point, at once c cosec is (211) z, In the present refers to C. R., and the lower to C. L. however, the positions of the instrument are less ambiguously expressed by circle west (C. W.) and circle east (C. E.), respectively. We may now use (33) to determine the hour angle of the star when its azimuth is equal to^ s in which the upper sign case, . Replacing A in (33) by A s and writing we may do since both are very small A and t / we angles, A whence by s instead of their sines, which find s\nz (212) (211) /cos d = asin z -f-cos2 c. (213) = 0, on account of the Equations (211) and (212) become indeterminate for .3 presence of A, but the conditions of the problem show that there can be no such discontinuity in the expression which gives / as a function of a, i>, and c. o, and becomes inapplicable only for Equation (213) is therefore valid for.s stars very near the pole. Since the star is near the meridian at the instant of i>. observation, z in (213) may be replaced by the meridian zenith distance <p = Writing A = s\n(<p and substituting d)sec3, for t in (210) we B = cos(<p 3) sec 8, C=secS, (214) find -bBcC. (215) PRACTICAL ASTRONOMY 122 the time of transit 6' Equations (214) and (215) give the value of J0 when across the vertical thread has been observed, provided the instrumental conThe quantities A, B, and C are called the stants a, b, and c are known. transit factors. Their values depend only upon the position of the star and, for any given latitude, may be tabulated with d as argument. They may also be tabulated with the double argument d and z. Tables of the latter sort are found in Kept. Supt. U. S. Coast and Geodetic Survey 1897-8, pp. 308-319. These are applicable for all points of observation. There remains still the determination of the constants, a, b, and c. The second of these can be made equal to zero by a careful adjustment and levellevel is ling of the instrument, or its value may be measured in case a striding The azimuth and collimation constants are best determined from available. the observations themselves. Assuming that b has been made equal to zero, or that its value has been determined, there remain in (215) only three unknowns, Ad, a, and c. The observation of any three stars will afford three equations of condition involving these quantities from which, theoretically, their values may be determined. Practically, however, the solution is simplified and rendered more precise by proceeding as follows: Suppose that the transits of a number of stars of various declinations have been observed, the instrument having been used in both positions. Consider the results for two of these having the same declination as nearly as possible, one observed C. W., the other, C. E. Writing to be J0' we have from = 0' + 5, (216) (215) zld = M' w + aA + cCw v ., M = M',+aA,cC, Since pose it is assumed that the two declinations are nearly equal, we may supwhence we find Aw = AR , J0' w '= J0'rE +r L-E (2I7) ' ~r **w which determines the collimation constant. Should there be more than one pair of stars of equal declination, (217) may be applied to each. the resulting values of c will then be accepted as the final value. Next, consider two stars observed whose declinations northern star, a differ as in widely circumpolar preferably, the other, a southern find for these objects from (215) of the same position of the instrument One of these should be a as possible. Jd"=J6'cC we The mean star. Writing (218) THE MERIDIAN METHOD = M\ + aA M = J0" + aA 40 s 123 H s whence ._45=j. "N (2I9) -"S Inasmuch as there the reversal, a should instrument. The value of Jd is danger of a change in the azimuth constant during be determined by (219) for both positions of the is then to be calculated by Jd = Jd" + aA. (220) The mean of all such values is the final value of the clock correction. The chief advantage of the meridian method of time determination is to be found in the fact that the results do not depend upon a reading of the circles. Since the uncertainty of an observed transit is considerably less than that of an angle measured with a graduated circle, the precision is relatively It is the standard method of determining time in observatories. high. When carried out with a large and stable instrument mounted permanently in the plane of the meridian, with the inclusion of certain refinements not considered in the preceding sections, it affords results not surpassed by those of any other method, either precision or in the in amount of labor involved in the reductions. 83. Procedure. To place the instrument in the meridian we may make use of a distant object of known azimuth. Set off the value of the azimuth on the horizontal circle and bring the object on the vertical thread by rotating on the lower motion, paving clamped the lower motion, rotate on the upper motion in until the reading is zero. The line of sight will then be approximately the plane of the meridian. In case no object of known azimuth is available, Polaris may be used inon the vertical thread at an instant for brought which its azimuth has previously been calculated by (184). With the exthat the must be made at a definite the instant, setting ception procedure is the same as that for a distant terrestrial object. The determination of the azimuth of Polaris requires a knowledge of the approximate time, but (191) shows that if 6 be known within two or three minutes, the azimuth will not be in error by more than one or two minutes of arc, which is sufficiently accurate. In case the clock correction is entirely unknown, an approximation may be derived as follows: Set on Polaris and clamp in azimuth. Then rotate the telescope on the horizontal axis and observe the transit across the vertical stead. In this case the star is thread of a southern star of small zenith distance. Denoting the sidereal clock time of transit by 0', the approximate error of the timepiece will be given by (221) PRACTICAL ASTRONOMT 124 Since the azimuth of Polaris differs but little from 180, the line of sight will not deviate greatly from the plane of the meridian, especially when directed toward points near the zenith. If the zenith distance of the time star is not more than 25 or 30 the error in d will not, ordinarily, exceed two or three of the azimuth minutes, and this, as stated above, is sufficient for the calculation instrument. of the of Polaris with the precision necessary for the orientation in each posifive stars The will include the observation of four or program Each tion of the instrument, reversal being made at the middle of the series. the of determination group should contain one northern star to be used for the azimuth constant. The remaining objects should be southern stars culminating In order that there may be preferably between the zenith and the equator. sufficient data for the determination of the collimation constant, care should be taken to observe at least one pair of stars, one C. W., the other, C. E., whose declinations are equal or nearly so. For an instrument whose vertical circle reads altitudes, the settings which will give the telescope the proper elevation to bring the stars into the field at the time of culmination are to be calculated by. Setting = 90 d), (<p (222) which the upper sign refers to northern stars. The star list with the setting for each object should be prepared in advance. This having been done, the instrument is to be levelled and adjusted Three or four minutes before the transit of the first star, which in azimuth. in 46, set the vertical circle at the proper readinto the field adjust in altitude until it moves along will occur at the clock time a ing, and as the star comes Note the time of the horizontal thread. its transit across the vertical thread After one-half the stars have been observed manner, reverse the instrument about the vertical axis through 180 to the nearest tenth of a second. in this and proceed with the observation of the remaining stars. Observations with the striding level for the determination of b should be made at frequent intervals throughout the observing program. Level readings increasing toward the east should be recorded as positive; toward the west, as negative. If a striding level is not available, the plate levels, especially the transverse level, should be very carefully adjusted before beginning the observations and the bubbles should be kept centered during the measures. The reduction is begun by collecting the right ascension, the declination, each star. The coordinates are to be interpolated for the instant of observation from the list of apparent places in the Ephemeris. The transit factors may be computed by (214), or better still, they may be interpolated from the transit factor tables. (See page 122.) If the inclination of and the transit factors for the horizontal axis has been measured, the values of b are to be computed by The value of Ad' is then to be calculated for each star by (216). Then (113). select two stars of equal or nearly equal declination, one observed C. W., the other C. E., and calculate c by (217). Compute as many such values of c as there are pairs of stars of equal declination, and form the mean of all. With the mean value of c calculate Jd" for each star by (218). Then determine a for THE MERIDIAN METHOD 125 each position of the instrument by (219), using for this purpose the stars of extreme northern and southern declination. Finally calculate Ad for each is the final value of the object by (220). The mean of all such values of clock correction corresponding to the mean of the observed clock times of M transit. In case the rate of the timepiece is large, each observed 6' should be cor- rected for rate before forming the values of JO', the corrections being applied in such a way that each 0' becomes what it would have been had all the observations been made reduced usually the exact hour or half-hour nearest the middle of the series. is Example 47. at the On same 1909, May 19, Laws Observatory was determined by & Buff engineer's transit. The error of the clock was The epoch instant. Wed. to which the values of 0' are P. M., the error of the Fauth sidereal clock of the the meridian method, the instrument used being a Buff known to be approximately -f- 7 m o'. The azimuth of Polaris h m 1 Vernier of the horizontal calculated by (184) for the clock time 5i o was 179 26'5circle was set at this value, and at the clock time indicated Polaris was brought to the inter- A n section of the threads by means of the lower motion. After clamping, the upper motion was released and vernier was made to read o. The instrument having thus been placed in the A meridian, the transits of four stars were observed. The reversal was then made by changing the reading of vernier A from o to 180, after which four more stars were observed. The plate levels were carefully adjusted throughout the observations. beginning, and the bubbles were kept centered at the The first of the tables gives the observing program and the data of observation. The various columns contain, respectively, the number, name, magnitude or brightness, and the apparent right ascension and declination of the stars; the setting of the vertical circle, the the observed clock time of transit, and the position of the circle. The settings were obtained by adding the colatitude 51 3' to the values of the declination. For northern stars this sum must be subtracted from 180. The second from each star. and the value of the clock correction derived by subtracting each #' from the correspondThe third and fourth columns contain the values of the table contains the reduction The values of jy are obtained ing a in accordance with (216). transit factors interpolated from the tables of the Laws Observatory. None of the pairs of stars observed are suitable for the determination of the collimation by (217). To avoid this difficulty, approximate values of the azimuth constant are derived by (219) from stars i and 4, and 6 and <ie 8, = + 4H. = J0" being replaced by J0' for this purpose. The results are \T -f-2!2 and These values are uncertain owing to the fact that the influence of the collimation has been neglected in deriving them, but they are sufficiently accurate for a determination of by (n6a), provided we use for this calculation stars whose declinations differ as little as possible. Substituting the numerical values of a, A, and C into (2i6a) for stars 3 and 5, and 2 c and 8 we find J0 = + 7 m 4M+J-4c J0 = -t-7 4.6 i.ooc J0 = + 7 m 4!4 J0 = -f 7 5-o + i-5c i.ooc sets of equations give for c, +0:10 and +0*27, respectively. The mean, 4-0*19, is accepted as the value of the collimation constant. Multiplying this by the value of C for each star gives the corrections for collimation contained in the fifth column. The combination of these with the value of J0' gives the quantities in the column headed J#". It should be noted that the algebraic sign of the collimation correction changes with the reversal of the These two instrument. The azimuth constant is now redetermined added to the values of J#" in for each position of the circle, using =+ and 6 and 8. The results are w 2*38 values of the azimuth corrections a A, which, accordance with (220), give J0, the clock correction for each star. for this purpose the value of JW" for stars I and a e -f4i6. From these we find the and 4, PRACTICAL ASTRONOMY 126 column contains the weight assigned to each result in forming the mean value of The mean J0 for the southern stars is the same for each position of the instrument, which shows that the influence of the collimation has been satisfactorily eliminated. It should be noted, as a control upon the calculation of the azimuth constant, that the values of 40 for each pair of azimuth stars must agree within one unit of the last The last the clock correction. place of decimals. No. In the present case the agreement is exact for both pairs. THE VERTICAL CIRCLE METHOD 127 but for those cases in which an uncertainty of one or two tenths of a second is permissible, the approximation is ample. In the meridian method both a and care determined from the observations themselves. Here we determine c as before, but a is to be calculated from the final result, (215) is insufficient; The azimuth constant will nearly equal the position of Polaris. azimuth of Polaris measured from the north point positive toward the east at the instant of setting, but not exactly, owing to the presence of the instru- known mental constants b and c. If a represent the azimuth of Polaris defined as above, we have by (82) and (89) a = a + b cot z a <p, , Since b and c are very small, za z being the zenith distance of Polaris. replaced by 90 c cosec may be whence a = a + b tan <pc sec Q (p. ( 22 3) Substituting (223) into (215) and writing B' = Atan<p+B, C = Asec<p + C, (224) we have 48 =a + a^ + WcC, 0' (225) where, as before, the upper sign refers to C. W. Equation (225) is the same in form as (215); but its solution is slightly different, for (184) gives # = it Gsecip sin/ , (226) which may be used for the calculation of This leaves in (225) only two unknowns, J0 and c, and the observation of any two time stars therefore affords the data necessary for a complete solution of the problem. For the sake of precision one of these should be observed C. W., the other, C. E. To determine c write . J0' We = 6' + a A+&B'. (227) then find from (225) whence f in Jg-.-Jg', f, * E i \ (~, W W (22S) There is here no necessity for an equality in declination of the two stars as the case of the meridian method, for the influence of the azimuth is in this PRACTICAL ASTRONOMT 128 Having found case included in Ad'. from (228) we calculate Ad from (225) c written in the form M=WcC. The factor accurately of a . A in (227) is the known than The as that in (215), but it must be more the meridian method, on account of the magnitude and are easily reduced by (214) to B' quantities in C B' in same (229) sec C = E + tan <p, (230) <p, which seed tan d. ( 2 30 values of E may be taken from Table X with d as argument, whence C' may itself be found by the simple addition of tan^>. For any given latitude with as be tabulated d The third column of Table X contains argument. may such a series of values for the latitude of the Laws Observatory, viz., 38 57'. The C The vertical circle method nation of time and azimuth. setting on and is easily adapted to a simultaneous determi- If the horizontal circle be read at the instant of addition, readings be taken on a mark, the azimuth of the mark will be given at once; for the azimuth of Polaris is calculated in the course of the reduction of the observations for time, and the horizontal Polaris, if in azimuth difference of the star and the mark. Since aa measured from the north point positive toward the east, the azimuth of the mark measured in the conventional manner will be circle readings give the is Am = MS + a i&o (232) M which S and are the means of the horizontal circle readings on the star and the mark, respectively; and am the mean of the calculated azimuths of in Polaris. The method, method of time determination, like that of the meridian not dependent upon the reading of graduated circles, and in conse- vertical circle is quence, yields results of a relatively high degree of precision. It possesses the further advantage that no preliminary adjustment in the plane of the meridian is necessary. It is especially valuable for use with unstable instruments, for the constancy of the quantities a, b, and c is assumed for only a very short interval, much less than in the meridian method. that the azimuth and level constants remain unchanged It is necessary only during the interval separating the setting on Polaris and the transit of the time star immediately following, and this need not exceed two or three minutes. Moreover, each set of two time stars is complete in itself and gives a complete determination of the error of the timepiece. The instrument used should be irregularity in the form of the pivots results. carefully constructed, however, for any is likely to produce serious errors in the THE VERTICAL CIRCLE METHOD 129 85. Procedure. The instrument is carefully levelled, and three or four minutes before the transit of a southern star across the vertical circle through Polaris, the telescope is turned to the north, and Polaris itself is brought to The instrument is the intersection of the vertical and horizontal threads. is noted. time of The teleand the sidereal azimuth in setting, clamped axis horizontal until its position is such that the rotated about is then scope , the southern or time star will pass through the field of view. The transit for a is observed, and the entire process is then repeated second time star, with the instrument in the reversed position. The data thus of the time star obtained constitute a set and permit a determination of the error of the timepiece. If a program simultaneous determination of time and azimuth be is required^ the for a set will mark and read the H. C. Set on Polaris, note the time, and read the H. C. \ C. Observe the transit of the time star. Set on Polaris, note the time, and read the H. C. ) Set on the W. V C. E. Observe the transit of the time star. Set on the mark and read the 11. C. which C. W. and C. E. are to be interpreted as meaning that if the instruto the north by rotating about the vertical will then be west or east, respectively. The plate levels the vertical circle axis, should be carefully watched, and if there is any evidence of creeping, the instrument should be relevelled. The observing list with the settings for the time stars should be prepared in advance. It is also desirable, in order to save time in observing and to avoid errors in the identification of the stars,' to calculate in advance the approximate times of transit. Disregarding the errors in level and collimation we have from (225) in ment be turned from the mark d' = a + a A-JO (233) which JO represents an approximate value of the clock correction. To determ aaA we combine equation (226) with the value of A from (214), and write in rive a value for the We thus find aaA=P(tand in tan <p), (234) 3O m /). (23?) \ J .// which > sin \o =47 ^ r Since a a A need be r (f). known only very i h w> roughly, for #, choosing for this purpose the sidereal mately to the middle of the observing program. 9 we may use a constant value time corresponding approxi- PRACTICAL ASTRONOMY 130 calculated ^01^(235)' we find the value of a A for each time value of d. One or two star from (234) by introducing the corresponding calculation. the for places of decimals are ample been secured, the first step in the reduction is The observations /having been having of sufficient the determination of an approximation for the clock correction errors in Polaris. of azimuth the of Neglecting calculation accuracy for the level and collimation we have from (225) M. = ad'+aaA, (236) which applied to the time star transiting nearest the zenith will give the For the term a A we may introduce the value calcurequired approximation. Collecting results we have the lated by (234) in preparing the observing list. following notation and formulae: , TT, and a, 8 are the coordinates of Polaris and the time star, respectively; and d', the sidereal clock times, respectively, of their observation; S and M, the readings of the horizontal circle for settings on Polaris and the mark, respectively; A m the azimuth of the mark rqeasured from the south, positive toward the west; Ad, the error of the timepiece, and J0 an approximation for this , , quantity. /= +^0 A = sin (?) (<p = , 4d' = a sec d, 0' C Log G G sec C C' = tan + E, <p + a A+bsec<f> e ~i L- (237) w be taken from Table IX, which is reprinted here for convenience, with the argument ta E or C\ from Table X with the argument d. The subscripts w and ^ refer to observations made circle west and or log tp is to ; circle east, respectively. Am = Y-z \M, Finally, calculate (S a.). + Mm (S OJ 180, (238) M refer to settings made with the instrument such a position that if turned toward the north by rotation about the vertical axis, the circle would then be west or east, respectively, according to the where the subscripts attached to in subscript. For the determination of the error of the clock, a should be expressed in seconds of time; for the determination of the azimuth, in minutes of arc. The values of A are needed to four places of decimals, and when once obtained, should be preserved, since, for a given latitude, they may be used unchanged THE VERTICAL CIRCLE METHOD 131 months. If the coHimation is known to be small and the declinations of the two time stars do not differ too greatly, it will be sufficient to for several take the mean of the values of Jd' for C. W. and C. E. as the error of the timepiece. TABLE <0 IX, 1910.0 TABLE X PRACTICAL ASTRONOMY 132 in minutes of arc. a whose logarithm is given in the fourth line is expressed the of time logarithm of 4, viz., 0.6020, Since the correction a. A must be expressed in seconds following it are added to two immediately the a and logarithms is also included when log is in satisfactory agreement with that form log a o A. The final value of the clock correction found in Ex. 47. The azimuth a Virginis, 1 sin/ ,, sec a <S 0' JO' C' cC' AO "o M Am C.W INDEX (THE NUMBERS REFEB TO PAGES. I Aberration: defined, 14; diurnal, Almucanters, Eccentricity: defined, 63; determination of for sextant, 90. 15. 9. measurement of with Altitude: defined, 10; engineer's transit, 77-80; Ecliptic, with sextant, 91-94. Altitude circles, mean 9. of, 27. axis of 1. relation to celestial 13; sphere, 7; defined, 10; calculation of from latitude, declination and zenith distance, 31-34; conditions for precise determina- Horizontal angles by repetitions, time, Hour 126-132. Azimuth and zenith distance transformed into hour angle and declination, 25-28; into right ascension and declination, 31. Calendar: Julian and Gregorian, 38-39. declination Hour angle and declination ; and zenith transformed and zenith distance, 29. defined, 24; axis of celestial fundamental for- mulae for, 32-34; conditions for precise determination of, 95; calculated from meridian zenith distance, 96; by Talcott's method, 97-99; from circummeridian altitudes, 99-102; from zenith distance at any hour angle, 102-103; from Collimation: error, 64; constant, 120. year, 39. Coordinates: necessity for, 8; primary and secondary, 9; systems of, 10; relative position of reference circles, 23-25; transformations of, 25-31. altitude of Polaris, 104-105; influence of an error in time upon, 106. Copernican system, 4, 6. Date: calendar and civil, 37. Day: apparent solar, 36; mean solar, 37; Leap year, 39. Least reading of vernier, 59-60. Level: error of, 64; theory of, 71-72; precepts for use of, 72; value of one divi- 37. 4. 80-81 sextant, 89. Clock: see timepieces. Coincident beats, 53. Diurnal motion, of, circles, 9. sphere, 7; of, 9; measurement Julian calendar, 39. Julian year, 39. Latitude: relation to 9S-102. 91. : Index error: of engineer's transit, 68; of Chronometer: see timepieces. Circummeridian altitudes: latitude from, Dip of horizon, of, 81-85. from latitude, distance, 31-32. Hour 7. sidereal, trigo- angle: defined, 10; transformed into ascension, 29-30; calculation of into azimuth 9. Declination: circles spherical right Celestial equator, 9. Celestial sphere: defined, 4; relation of its position to latitude, azimuth and time, Common of 91. influence of error in time on, 114-115; Cardinal points, 13. 51. nometry, 21-23. Gregorian calendar, 39. Horizon: defined, 8; artificial, 59; dip from the sun, 109-110; from from measured star, 110; zenith distance, 113-114; of mark, 114; with of, Fundamental formula; tion of, 108; determined precession Error of timepiece, clrcumpolar simultaneously condi- Ephemeris, 12. Equal altitudes: time from, 117, 118-120. Equation of time, 40. Equator: celestial, 9; mean, 13. Equinox: vernal and autumnal, 9; mean, Artificial horizon, 59. Azimuth: 61; historical, of horizontal angles, 80-85. 36; converted solar time, 40. Arguments: arrangement Asteroids, transit: tions satisfied by, 62-63; theory of, 64-71; measurement of vertical angles, 77-80; Apparent place, 16. Apparent solar time: defined, into 5, 9. Engineer's defined, 10. sion, 73-76; Longitude, Mean 133 constant, 120. 7. equator, 13. INDEX 134 Mean equinox, 13. Mean noon: defined, 37; sidereal time of, 47. Mean Mean Mean place, 16. solar day, 37. solar time: defined, 36; into apparent solar time, 40; converted converted defined, 36; right ascension of, 44-46. Meridian: defined, 8; reduction to, 100. Meridian method.^of time determination, 117, 120-126. Meridian zenith distance: latitude from, of, 2. Solstices, 9. fundamental for- mulae, 21-23. Standard time, Stars: motions 37. Stellar system, 1. of, 3, 15. 14, Successive approximations, 34, 118, 120. annual motion of, 5; parallax of Sun: 19-20. 96. Method Talcott's method, 97-99. Time: relation to celestial sphere, 7; fundamental formulae for, 32-34; basis of of repetitions, 81-85. Nadir, 8. Nebulae, 3. Noon: apparent, Parallactic Parallax: Planets: mean, 36; 37; sidereal, 38. 13. Nutation, 31. angle, defined, names, diameters, 2. Polar distance, Polaris: theory, 18-20. relative distances and from, 104-105; azimuth method of time de- vertical, 9. historical, of, discussion of, 58. True place, 16. True solar time: see apparent solar time. Vernier: of, 81-85. theory, 59-60; Vertical angles: 44-46. gineer's 31. 77, 93. 85; theory, re- 86-87; adjustments, 88-89; index correction, 89; 77-80; of with en- with sextant, 91-94. Year: tropical, common, Zenith, 4. measurement transit, Vertical circles, Right ascension and declination transformed into azimuth and zenith distance, historical, uncertainty of sults, 60-61. Residuals, 57. Semidiameter, and care Tropical year, 38. Right ascension: defined, 10; transformed into hour angle, 29-30; of mean sun, diurnal, of, see engineer's transit. Transit factors, 122. 16-18; table, 20; differential, 98. method error 52-58; 50-51; comparison Transit: Proper motion, 14, 15. Ptolemaic system, 4, 6. Rate of timepiece, 51. Reduction to the meridian, 100. 12; 48; solar, determining, 116-117; from zenith distance, 117-118; from equal altitudes, 118-120; meridian method, 120circle Polaris vertical method, 126; rate of, 51; defined, mean into be- solar into side- of Timepieces: Precession, 13, 15. Sextant: 47; relation 40; mean 126-132. termination, 117, 126-132. Poles of celestial sphere, 8. Rotation: 42; sidereal units, methods Polaris vertical circle Repetitions: and vice versa, tween real, 10. latitude Refraction: 36; different kinds, 36; distribution of, 38; difference in two local times, 39; apparent solar into mean measurement, solar 14; 1; from, 110-114. Prime into mean solar time, 48-49. Solar system: parts, 1; model Spherical trigonometry: into sidereal time, 47-48. Mean sun: eccentricity, 90; precepts for use of, 91. Sidereal day, 37. Sidereal noon, 38. Sidereal time: defined, 25, 37; converted 8. 38; Julian, 39; leap and 39. 8. Zenith distance: defined, 10; latitude from, azimuth from, 113; time 96, 102-103; from, 117-118. Zero reading, 89. 01KOH01 dO AIISH3AINn

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