ASTRONOMY FOR ENGINEERS PRACTICAL SHARES
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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