A COMPARISON of the METHODS USED in DETERMINING AZIMUTH by SOLAR OBSERVATIONS by .Gerald E. Murphy a Thesis Submitted to The

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A comparison of the methods used in determining azimuth by solar observations

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Authors Murphy, Gerald Edward, 1931-

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Link to Item http://hdl.handle.net/10150/319786 A COMPARISON OF THE METHODS USED IN DETERMINING

AZIMUTH BY SOLAR OBSERVATIONS

. Gerald E. Murphy

A Thesis Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE

In the Graduate College

THE UNIVERSITY OF ARIZONA

1964 STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of re quirements for an advanced degree at The University of Arizona and is deposited in The University Library to be made available to bor rowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in their judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: ^

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

^PHlIlP B. NEWlAN Date Associate Professor of Civil Engineering TABLE OF CONTENTS

Page

LIST OF PLATES oeoeooeeeeoooeeaoooeQooooeeeooooeoeeeetooooeeoeoeG IV

LIST OF I LLUSTRATI ONS ©eoooooefroeeGeeooeoooeoeeGeoseooooec-oocooe V

INTRODUCTION o©ee©e«e»»t»©©oo©e®

CHAPTER I

THE HISTORY, ADVANTAGES, AND DISADVANTAGES OF SOLAR OBSERVATIONS

The History of Solar Observations »e»,,,*co 3 Advantages 00G0etieeo0ee®©6000e9000tf0»000»e000eGe000@e»oe0e-g©cc»0 V Disadvantages e©o®epeoe6oc<8©e6ooeoso6oo6©9oee®e©eo©©»©

CHAPTER II

DETERMINATION OF AZIMUTH BY SOLAR OBSERVATION

DefimtlOnS and Notations oocee©o»ebeo©»eeo»©©©e©e©o©eeoo«>ye©©ee XX The AStrOnOIDlCal Triangle o6 0 ©©oo©eGOOoeoooeaooo©o©eeeo«ee©oo©e© X^

B aS 1 0 Equati ons »e©e»et»eeotiooeooociooeooeee©e»oeoooooo©»o©©»

CHAPTER III

METHODS OF OBSERVING THE SUN

Solar Screen ooco©©eoceD»o»©ooeo©oe©©6 esooeoo©©e©oeoeeaeooGo 6 oee 17

Solar Filter eoeeeo©oo6o©©®ee»o6ooe©o®©o©©o©oo©&ooe©o©ooo©o©oece 2 4

S O la r Re tic le G®©o©»eooeot>o©»oco»©©e®o»ooeooos©©©o©eeeoeo<»ooao®e 20

Simplex Sdar Shield ©o©oeoe»eooooe©oooe»©»eee©eooeee©«co0©c6o©o 27 RO0 IpfS Sdar Prism eeoeeooo©e©ee»oo6<>©©®oooeeeoocQ"eo»c©©©ci

111 iv CHAPTER IV

CORRECTIONS FOR THE SUN’S CURVATURE AND SEMIDIAMETER ■ Page S e ITil c3.1 ame *t elP oeeeoQeooeoeeoeooeoesoooe. ee^tieoeoeoeoooeoeeeecoeeeo-- 31 Curvature Correction ©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©oo©©©©©©©©©©© 34 Semidiameter Correction ©©©©©©©©©©©©©©©©o©©©©©©©*©©©©©©©©©©©©©©© 36

CHAPTER V

DETERMINING DECLINATION, LATITUDE, AND LONGITUDE

Declination ©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©o©©©©©*©©©©©©© 42 Latitude ©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©o©©©©©©©©©©©©©©©©©©©© 45 Longitude ©©o©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©© 46

CHAPTER VI

AZIMUTH BY THE ALTITUDE OF THE SUN

Introduction ©©©©©©©©©©©©©©©©©©©©©©©©©©©o©©©©©©©©©©©©©#©©©©©© 51 Trigonometric Formulas ©©«©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©© 51

Factors Affecting the Measuring of Altitude ©lootiooeeeeeceeooo 54 Effect of Errors in Altitude on the Computed Azimuth © © 65 Effect of Errors in Declination on the Computed Azimuth 67 Effect of Errors in Latitude on the Computed Azimuth ©©©©.©©©© 71 Field Procedure for Observations 73

CHAPTER VII

AZIMUTH BY THE HOUR ANGLE OF THE SUN

Introduction ©©@©©©©©©©©©©©©*©©©©©©©©©©©**©©©©*©©©©©©©@©©©©@©0 ©© 76 Determining the Hour Angle of the Sun .©...©.©©..«..© .©©»»©©.©. © 78 Factors Affecting the Measurement of the Sun's Hour Angle 81 Field Procedure for Observations 85 CHAPTER VIII

OTHER METHODS OF DETERMINING AZIMUTH BY THE SUN

Azimuth by the Altitude and Hour Angle of the Sun

Ec^ual Altitude Method e>©o©®®G©oooo©®9 oo©ooooo©ocfc

CHAPTER IX

COMPUTATIONS

Introduction o©©©©©©©©©©©©©©©©©©©©©©©®©©©©©©©©©©© Slide Rule o©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©© Logarithms ©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©o©©©©©©© Natural Functions ooeooo©©©©©©©®©©©©©©®©©©©©©©©©©

Electronic Digital Computer 006606006690

Hour Angle Program 606000000006600 0 060660006060

CHAPTER X

CONCLUSION

C O n d U S l O n ooooooooeeoeoeooooooeoeeooeooeoeooeoo

BIBLIOGRAPHY oeoOooceooooooooooeooosadoaooooeeooo LIST OF PLATES

Plate Page

1 Computation of Sun9s Declination 44

2 Telescopic Solar Declination Setting e*,*,*.*.,..46

3 Determination of Azimuth by Log Secants „»* * *„*»,»*,* * * a 55

4 Alt ltude« Azimuth Curves »ec>»©ooo6 eot>oooo6 oeoeeoeoeoooo6 » 68

5 Effect of An Error In Declination on The Sun9s Hearing efeoooo&oooooefcQCtoO’eefreoooooeoeeeeoooeoooooo 70

6 Effect of Latitude and Hour Angle on dB/dPhi © „o* . 72

7 Altitude«-Azimuth Curves With dB/dPhi© Curves Superimposed ©©©eo©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©©© 74

8 Error In Azimuth For One Second Error In Time ©»© © © © © © © © 86

vi LIST OF ILLUSTRATIONS

Figure Page

1 Groma ...... roooeeooooeeoooooooeotiotieoooeteooooeooooeoo 4

2 Indian Circle , ofroeeeoooeoetooooeq&oeeoooeeoooGQQOoooooo 5

3 Celestial Triangle ...... eeeoeoeeeeooooooyooeoefrooooo „ 12

4 Spherical Triangle 'Oooooeeeooooo 6e>eepoeo©6e©,oooedooeo' . 14

5 Quadrant-Tangent Method eeeeeooooooeoooooeoi y o o o o o e e 19

6 Center-Tangent Method eeeoooeeooooooOeeoeooeoeo&eceoooeoo 21

7 Bisect! On eeooeeeeeooeeooeeeco. eoooooceeeoeeoeeoeeoeeceeoeo 23

8 Solar Filters Quadrant-Tangent Method eoeoeeeeoodeofloooeo 25

9 Solar Filters Center-Tangent Method , o 0 © o o o 25

10 Solar Reticle odeeeeeeoeeoeoee'ea ©eeoeooooooooeoeoeoeee.tii 26

11 Roelofs9 Solar Prism i e © e e © o e 1 ©©ooBooaoooooeoooooooo 29

12 Semidiameter Correction AooeoooeeeooeoBoeoooooi 31

13 Horizontal Angle Correction for Sun 9s Angular Semidiameter „*,* * *,„ oooooeooooao 33

14 Effect of Curvature •• oeeeooeooo 35

15 Effect of Semidiameter eoccoooc^o oooooooqooo e>© ooo 37

16 Quadrant-Tangent ©oeeeeooeoeefeaece oeeeooeooeoooooeooeoooo 39

17 Accuracy of Measuring Latitude « 47

18 Accuracy of Measuring Longitude ooooeooe 49

19 XnStrUmeTit Error eeooooooooooo&doeooooosooeeeooooooooooool 57

20 Refraction Q © © © s © © o ©«© © © © o © ooooocoeeoooeoooeooooooooe 60

vii viii

Figure ' Page

2 3L P erS i 3-SX. o*eeoode»ooeoeeoeooeoo»Qoo»eoei?oeeootioo<)oeo@o#eeo 03

22 Relationship Between the Greenwich Hour Angle

an (1 Local Hour Angle oceoeeeeooeoooeeoeotieeooeoeooooe 61 A COMPARISON OF THE METHODS USED IN DETERMINING

AZIMUTH BY SOLAR OBSERVATIONS

Gerald E. Murphy

Abstract

A number of methods are available to determine azimuth by the sun. The basic equations are derived and each method examined. Devices for pointing on the sun are discussed in detail. All factors that are related to the computed azimuth such as declination, latitude, and longitude are considered along with the measurement of each. The altitude and hour angle methods are compared and the factors affecting the accuracy of each method discussed. The solar equations can be solved in a number of ways. The standard methods of performing the computations are reviewed and a digital computer program to solve the hour angle

.equation is presented.

ix INTRODUCTION

The science of surveying had its birth at the time man first recognized the right of private ownership of land* This recognition was impossible without boundaries9 no matter how crude* to delineate one manffs holdings from his neighbors* As populations increased and

land became more valuable the status of the land surveyor grew* In

540 A * D * Cassiodorus wrote the following concerning the place of the

land surveyor in Roman life*

The professors of this science [of land surveying] are honored with a most earnest attention than falls to the lot of any other philosophers* Arithmetic9 theoretical geometry9 astronomy9 and music are discoursed upon to listless audiences9 sometimes empty benches6 But the land surveyor is like a judges the deserted fields become his forum9 crowded with eager spectators* You would fancy him a madman when you see him walking along the most devious paths* But in truth he is seeking for the traces of lost facts in rough woods and thickets* He walks not as other men walk* His path is the book from which he reads % he shows what he is saying; he proves what he hath learned; by his steps he divides the rights of hostile claimants; and like a mighty river he takes away the fields of one side to deposit them on the other,^

The measurement of distance and the determination of direc tion were an essential part of these early surveys* It seems highly possible that the lofty position held by the land surveyor in early

^-Edmond R* Kiely* Surveying Instruments: Their History and Classroom Useg Bureau of Publications* Teachers College* Columbia University$ New York, 1947, page 43*

1 Roman times was partly due to the skill he had developed in determin ing direction. The use of the sun and stars to establish the meridian surely impressed the landowners of Rome,

The present day land surveyor is still called upon to determine the true bearing of boundary lines. Most of our state and county boundaries9 and all surveys of the public lands of the United States since 18559 are defined in terms of astronomic or true north$ Celes tial observations are the only means of establishing a true meridian.

The following chapters compare the methods used in determining this meridian by observations on the sun. CHAPTER I

THE HISTORY„ ADVANTAGES„ AND

DISADVANTAGES OF SOLAR OBSERVATIONS

The History of Solar Observations

The practice of determining azimuth by the sun can be traced

to the earliest Etruscan historyc In the sixth century B.C, these people established the meridian by the rising and setting of the srnn

The method consisted of staking the east-west line e the decimanus, by

observing the sun at its first appearance in the morning and again as

it set in the afternoon. The north-south line, the cardo, was established by means of the groma» A sketch of the groma is shown in

Fig, 1,

The groma was used to construct a perpendicular to the decim

anus, The Etruscans were aware of the fact that this method gave the

true meridian only at the time of the equinoxes, In their writings

they recommend that the decimanus be established only from the shadow

of the sixth hour.^

Another early method of establishing the meridian was known

as the Indian Circle Method. The. oldest Indian description of this

method is to be found in the Surya Siddhanta., an astronomical work

j-Ibid., page 32 ■

3 4 dating from about 400 B.C.^

Fig. 1.— Groma

The method was as follows: A circle was drawn on a carefully leveled section of ground, A vertical rod was placed at the exact

^Ibid., page 61 center. When the extremity of the stick's shadow touched the circle both in the morning and afternoon a point was marked. Fig. 2 shows the

Indian Circle with straight lines connecting the two points and the

center of the circle.

Fig. 2.— Indian Circle

The bisection of the lines formed was the true meridian.

Bisection was accomplished solely by means of measured distances in

much the same manner as described in present day surveying textbooks.

The Romans adopted the Indian Circle method and it was used extensively for the establishment of street lines and the determination

of boundaries.

The Moslems introduced various refinements to the Indian

Circle method. One important contribution was the use of a number of

concentric circles instead of one. This made it possible to mark a number of points on different circles in both the morning and after noon. Better accuracy was obtained and the meridian could still be

determined even if clouds obscured the sun for a portion of the day.

The determination of azimuth was important to the Moslems both in

their astronomical work and in determining the direction of Mecca for

some of their religious rituals.

The accuracy of the Indian Circle method can be more appreciated when compared with the magnetic compass. The earliest records of

scientific observations on the variation of magnetic declination were

made by Felipe Guillen in 1525,*^ JBy means of the Indian Circle

method he determined the meridian and then found the angle between the

compass needle and the meridian.

The first tables of the sun*s declination were published in

Hebrew in 1473 by Abraham Zacuto, They were later translated into

Latin by Jose Vizinto, These tables and other useful information to

mariners were printed in manuals or regimentos. They were used to

calculate latitude by meridian altitude observations of the sun.

The early property surveys in the New Wprld were made with

instruments and methods little better than those used in ancient times.

Direction was usually determined by magnetic compass. The variation

in magnetic declination was checked by observations of Polaris at

elongation,

LIbide * page 214, The rectangular surveying system used in the United States

is dependent on the establishing of cardinal courses for controlling

lines„ The early surveys of the public land were dependent on the needle compass for direction* The compass was first referenced to magnetic north and the magnetic declination was then turned off* As

the surveys progressed into the upper regions of the Great Lakes the magnetic compass proved so erratic that its use had to be discontinued*

The Burt solar compass was introduced in Northern Michigan about 1836o*^ The solar unit, was later mounted on the telescope of a

transit* The use of solar observations proved both reasonably

accurate and relatively inexpensive in the survey of the public domain*

The development of optical transits and the improvement of

methods of pointing on the sun have made solar observations an im portant part of modern surveys*

Advantages

The surveyor who rises at some darks frigid hour to observe

Polaris at elongation can readily understand one advantage of deter

mining azimuth by solar observation* Solar shots are taken only in

the daytime and usually during regular working hours* Avoiding a

special trip to the field for the sole purpose of taking a star shot

^The Gurley Telescopic Solar Transit: Its Use and Adjustments9 Bulletin No. 112-T* W. 6 L* E. Gurley, Troy* New York* page 15* could mean a substantial savings in the cost of a surveye

Working in daylight hours offers several advantages» In remote areas and in rugged terrain9 it is much easier to find the instrument station9 and there is no need to occupy the target station for the purpose of illuminating the target* Instrument setups are faster % and there is less danger of accidently disturbing the transit by bumping a tripod lege Mistakes in reading and recording both angles and time are fewer than when working in semi~or total darkness*

Solar observations can be successfully made when observing conditions are relatively poor. Partly cloudy and hazy conditions that would prevent a Polaris shot seldom interfere with a solar observation.

In northern latitudes9 observations of Polaris, require rel atively large vertical angles in comparison with solar shots* This will result in proportionally large errors in vertical and horizontal angles if an instrument is in poor adjustment. The error in the horizontal angle for a one minute inclination of the vertical axis is four times as great for a vertical angle of forty-five degrees as compared with a vertical angle of fifteen degrees

^Raymond E. Davis and Francis S e Foote9 Surveying: Theory and Practice* McGraw-Hill Book Company9 Inc,s New York, fourth editions 19539 page 305. Disadvantages

The size and brightness of the sun are probably the major drawbacks in solar observations* Its large size makes it difficult to point at its exact center* Several methods of pointing and numerous devices have been used 9 but none so simple or accurate as pointing on a star* The extreme brightness of the sun requires the use of a darkener or solar screen to prevent serious damage to the observer8s eye*

The computations required to determine azimuth by the sun are longer and more complicated than those used in reducing a Polaris observation* An error in computing the bearing of Polaris will result in an azimuth that at most is wrong by two or three degrees, whereas an error in computing the bearing of the sun may give an absurd answer*

Parallax is an added correction that must be considered on solar shots that can be ignored on other stellar observations*

Several other factors that affect any survey work performed in direct sunlight should be mentioned* "Surface wind speed is usually at a minimum about sunrise and increases to a maximum in early afternoon * Atmospheric conditions are much more variable and result'in unequal atmospheric refraction* The sun8s rays often strike one side of the instrument while the other side remains

*^Ray K* Linsley% Max A* Kohler9 and Joseph L* H* Paulhus9 Hydrology For Engineers, McGraw-Hill Book Company* Inc,, New York, 1958, page 21/ 10 shaded* This results in a temperature difference and unequal expansion of parts of the telescope*

Conditions most favorable for the precise measurement of angles are in direct conflict with the requirements for solar observations*

The use of umbrellas* overcast skies* or nighttime observations are impossible when determining azimuth by the sun* CHAPTER II

DETERMINATION OF AZIMUTH BY

SOLAR OBSERVATIONS

Definitions and Notations

The determination of azimuth by solar observation requires that a system of coordinates be used that will enable the surveyor to compute the bearing of the sun* A number of astronomical systems of coordinates are in common use at the present time» Only two of theses the horizon system and the equator system9 are needed in determining azimuth by solar observation0 Fig. 3 shows the celestial triangle5 both in the horizon and equator system. To understand the celestial triangle, a number of terms must be defined in each system and their notation given.

The Horizon System

Celestial Spheres An imaginary sphere of infinite radius with its center at the center of the earth.

Sun (S): The star nearest the earth about which the earth revolves«,

Zenith (Z): A point directly overhead. This would be the point at which a plumb line projected upward would pierce the

celestial sphere.

Nadir: The point directly opposite the zenith on the

celestial sphere.

11 12

o "

Fig. 3.— Celestial Triangle

Horizon: A great circle defining the intersection of the celestial sphere and a plane perpendicular to the line joining the zenith and nadir and halfway between these points.

Vertical Circles: Great circles passing through the zenith and nadir.

Meridian: The vertical circle which passes through the celestial poles.

Azimuth (B): The azimuth of the sun is the angle at the zenith measured eastward or westward from the meridian to the verti cal circle through the sun. Altitude (h): The sunes angular distance above the horizon,

It is measured upward on the vertical circle through the sun from the

horizon to the sun®

Zenith Distance (z)i 90°«h

The Equator System

Celestial Equator: The intersection of the plane of the earth9s equator with the celestial sphere«>

Celestial Poles (P): The two points where the axis of ro

tation of the earth extended pierces the celestial sphere®

Hour Circle: Great circles perpendicular to the equator and passing through the celestial poles.

Meridian: The hour circle through the zenith of an observer.

Hour Angle (t): The angle at the pole from the meridian westward to the hour circle through the sun.

Declination (Dec.): The angular distance measured on the hour circle through the sun from the equator to the sun. It is positive when measured northward from the equator and negative when measured southward.

Polar Distance: 90o-Dec»

Latitude (Phi*): The angular distance of the observer north

or south of the equator measured along a meridian of longitude0

Longitude: Angular distance of a meridian east or west of a

starting meridian through Greenwich^ England& measured along the

equator* 14

The Astronomical Triangle

The astronomical triangle, like any spherical triangle, has certain relationships between its sides and its angles. These laws are derived in any text on spherical trigonometry and the derivation will not be repeated. Referring to Fig. 4.(a) the three most important formulas in the solution of any spherical triangle are:

(a) (b)

Fig. 4.— Spherical Triangle

The law of sines.

sin A _ sin B _ sin C ( sin a sin b sin c

The law of cosines.

cos b = cos a x cos c + sin a x sin c x cos B ...... (2)

Relationship between two angles and three sides.

sin a cos B = sin c x cos b - cos c x sin b x cos A (3) 15

The application of these three laws to the astronomical triangle results in the basic equations for determining azimuth.

Basic Equations

The astronomical triangle is shown in Fig. 4.(b) and equations

(1), (2), and (3) applied.

The law of sines.

sin B _ sin t sin (90°-Dec.!) sin (90o-h)

Sln B = cos Dec. x sin t /ltX(1,)

The law of cosines solving for B.

cos (90°-Dec.) = cos (90°-h) x cos (90°-Phi.)

+ sin (90°-h) x sin (90°-Phi.) x cos B

After reduction this equation becomes

sin Dec. = sin h x sin Phi. + cos h x cos Phi. x cos B

D _ sin Dec. - sin h x sin Phi. ,c\ COS B - ■ ...... Ko) cos h x cos Phi.

The law of cosines solving for t.

cos (90°-h) = cos (90°-Phi.) x cos (90°-Dec.)

+ sin (90°-Phi,) x sin (90°-Dec.) x cos t

After reduction this becomes 16

sin h = sin Phi. x sin Dec. + cos Phi. x cos Dec. x cos t

+. _ sin h - sin Phi. x sin Dec. COS L w ■ — 1*1 I w n > T ■! n I I ■ 1 m - cos Phi. x cos Dec,

Relationship between two angles and three sides.

sin (90°-h) x cos B = sin (90°-Phi.) x cos (90°-Dec.)

- cos (90°-Phi.) x sin (90°-Dec.) x cos t

After reduction this becomes

cos h x cos B = cos Phi. x sin Dec.

- sin Phi. x cos Dec. x cos t ...... (7)

Another useful equation can be developed by dividing Eq. (7) by Ec . (4).

cos h x cos B _ cos h x sin B

cos Phi, x sin Dec. - sin Phi, x cos Dec. x cos t cos Dec. x sin t

cos Phi. x tan Dec. - sin Phi. x cos t cot B ------JJ— ------CHAPTER III

METHODS OF OBSERVING THE SUN

Solar Screen >r

Viewing the sun directly through a telescope may result in serious injury to the observer's eye. There are several safe ways of pointing a telescope at the sun. The solar screen is probably the most commonly used method of viewing the sun. The observation can be made with any transit that contains a vertical limb. The screen may consist of a special attachment supplied by the instrument manufacturer9 but more often it is simply a white card held by hand a few inches back of the eyepiece® The advantage of the attach ment is that the screen remains perpendicular to the axis of the telescope and at a fixed distance from the eyepiece. In either case the screen is used to reflect the sun*s image.

The sun is sighted by pointing the telescope in the direction of the sun and observing the shadows cast by the telescope vial posts* When the shadows appear to coincide the azimuth motion is locked. The telescope is then rotated about its horizontal axis until the image of the sun flashes across the screen® The vertical motion is then locked. By carefully focusing both the objective lens and the eyepiece* the shadows of the cross wires are visible against a sharp image of the sun,

i. 18

When using a solar screen there is still the problem of pointing at the center of the sun* The pointing can be accomplished by any of the following methods«

Quadrant-Tangent Method

The sun9s image is brought into view on the solar screen in such a position that it is tangential to both the horizontal and vertical cross wires* At the moment of tangency the time 9 vertical anglef and horizontal angle are read and recorded* Knowing the semi diameter of the sun the correct vertical and horizontal angle to the sun9s center can be computed*

An easier method of obtaining the correct vertical and hori zontal angle to the sun*s center requires taking observations in pairs* The second pointing places the sun9s image in a diagonally opposed quadrant from the first* Assuming the path of the sun is a straightline between pointings9 the mean of the-vertical and hori zontal angles requires no correction.

Since the sun is moving rapidly in both altitude and azimuth 9 it is difficult to follow it by manipulating both motions of a transit* Sighting can be simplified by selecting a quadrant in which the sun9s image is moving toward one cross hair and away from the other. One wire is set to cut a segment of the sun*s image that is moving away from the cross hair® This wire is then kept stationary while the sun is tracked with the other wire* The instant the edge of the sun becomes tangent to the stationary wire, all motion is stopped* By this method a simultaneous tangency can be obtained* 19

The correct quadrant to place the image of the sun is dependent on when the observation is made and the type of telescope.

Fig. 5 shows the image of the sun as it appears on a solar screen with an erecting telescope.

A.M. P.MP.

Hor. Wire Vert. Wire Hor. Wire Vert, Wire Stationary Stationary Stationary Stationary

Fig. 5.— Quadrant-Tangent Method

When an inverting telescope is used Fig. 5. should be turned upside down.

A number of different procedures are used when taking a series of observations by the quadrant-tangent method. The K 6 E Solar

Ephemeris recommends taking at least three successive readings with the sun's image in the same quadrant before the telescope is reversed.

An equal number of pointings are then made in the diagonally opposed quadrant. The averages of the times, vertical angles, and horizontal angles are used to compute the bearing. This method is subject to rather large curvature errors since a time lapse of ten to fifteen 20 minutes may occur between the initial and final sighting*

A more refined method of observing is used by the Bureau of

Land Management* They treat each pair of observations as a series and from the average readings compute an azimuth* A total of three sets are taken and the average value of the three computed azimuths used as the true bearing*

The Geological Survey recommends a total of ten observations*

Five sightings are taken with the sun in the same quadrant* Upon completion of the fifth pointing the initial station is sighted* The telescope is then reversed and five pointings taken in the diagonally opposed quadrante The mean values of the ten pointings are used in computing azimuth. The Geological Surveyb method of reading horizon tal angles differs from most procedures. The A vernier is read for all pointings with telescope direct and the B vernier for all pointings with the telescope reversede This method enables the observer to read the angles faster, and there is less danger of accidently bumping a tripod leg* A time limitation of ten minutes is used between the first and last pointing.

When using the average values of several pointings taken over a period of time, some error is introduced* The effect of assuming the stings path is a straightline is discussed in Chapter IV*

Center-Tangent Method

The quadrant-tangent method involves watching two points of tangency at the same time* Since the points are approximately sixteen minutes apart it is impossible to observe them simultaneouslye 21

The best an observer can do is to view them alternately as rapidly as possible. The center-tangent method overcomes this difficulty by requiring only one point of tangency to be observed.

The image of the sun is brought into view on the screen in the same manner as in the quadrant-tangent method. One wire is kept centered on the sun while the other wire remains stationary and allows the sun's image to make its own point of tangency. The moving wire is kept centered on the sun by bisecting the small and diminishing segment. The sun's image, as it appears on a solar screen with an erecting telescope, is shown in Fig. 6. © A.M. ® © P.M. Bor, Wire Vert. Wire Hor. Wire Vert. Wire Stationary Stationary Stationary Stationary

Fig. 6.— Center-Tangent Method

When an inverted telescope is used Fig. 6. should be turned

upside down. Use of the center-tangent method requires that each obser vation be corrected for semidiameter» The sun *3 semidiameter is always added to the vertical angle for observations taken in the A.M. and subtracted from observations taken in the P»MC The semidiameter correction to horizontal angles is.equal to the sun*s semidiameter9 as given in the ephemeris9 multiplied by the secant of the sun5s altitude* Confusion as to when the correction should be added and when it should be subtracted from the observed horizontal angles can be avoided by use of the following rule*

No matter in what manner the observation is made 9 with inverted telescope9 prismatic eyepiece9 or image projected on paper* the eastern limb is always observed when the disk of the sun appears to leave the vertical wire* This will cause the correction for semidiameter s , always to be added to the horizontal angle reading on the sun k 9 provided the angles are measured in a clockwise direction *

The Gurley Ephemeris recommends at least three successive readings should be taken with the vertical wire stationary and the horizontal wire in movement* then an equal number with the horizontal wire stationary and the vertical wire in movement * The telescope is then reversed and a second set of six observations made in the same manner* The average of each set of readings is found$ then the average of the two averages used to compute the azimuth*

The above procedure of using the average readings taken over < period of time is subject to curvature error* Chapter IV deals with

“klason John Nasau* A Textbook of Practical Astronomy* McGraw- Hill Book Company* Inc* 9 IsWT^age^l43o 23 the magnitude of this error and methods of correcting for it.

Bisection

The simplest method of sighting the sun is to bisact the sun's image with both the horizontal and vertical cross hairs. Fig. 7 shows the sun's image when correctly bisected.

Fig, 7.— Bisection

This method of pointing is probably the least usei but offers these advantages:

There is no confusion as to which quadrant the son's image should be placed.

No correction necessary for semidiameter.

The time required for pointing is less than when osing the center-tangent or quadrant-tangent method.

The major disadvantage is the difficulty in pointing on the center of the sun. Philip Inch, writing in an ASCE transaction, con sidered the method of bisection accurate enough to give rasults within 24

one minute of azimuth0 He wrote the following concerning the problem

of pointing.on the sun5s center:

This (pointing) is best done by considering the intersection of the cross hairss as a point and placing this point in the center of the sun*s image» The human eye can place a point in the center of a circle with considerable accuracy % as witness the principle of the rifle peep sight*1

Like the quadrant-tangent and center-tangent methods9 there

is little agreement as to the observing procedure* The only consis

tent requirement is that an equal number of observations be taken in

the direct and inverted positions of the telescopee

Solar Filter

The solar filter or darkener is simply a colored glass that

is attached to the eyepiece of a transit* The use of a filter per

mits direct viewing of the sun without danger of injuring the eye.

For high angles of observation the filter can be used in connection with a diagonal prism.

The filter must be attached.in such a manner that it is easily

movable. This permits the observer to sight a ground target, move

the filter into position in front of the eyepiecee and then sight the

sun with little delay.

The observing procedures and methods of sighting on the sun

are the same as when using a solar screen. Figs$ 8 and 9 show the

^Philip L, Inch* Simplified Method of Determining True Bearing Transaction of ASCE, Vol. 102, 1937, page 970. 25 sun as viewed directly through an erecting telescope.

A.M. P.M.

Hor. W i re Vert. Wire Hor, Wire Vert. Wire Stationary Stationary Stationary Stationary

Fig. 8.— Solar Filter: Quadrant-Tangent Method

A.M. P.P

Hor. Wire Vert. Wire Hor. Wire Vert. Wire Stationary Stationary Stationary Stationary

Fig, 9.— Solar Filter: Center-Tangent Method 26

Solar Reticle!

The precision of sighting the sun's center, either by use of a solar screen or a darkener, can be increased by use of a solar reticle.

A solar reticle is similar to the reticle in any transit in that it contains a vertical and horizontal wire. It also contains stadia hairs and is manufactured with a stadia ratio of either 1:100 or 1:132. The added feature is that it contains a solar circle equal to the image of the sun's diameter. A solar reticle with a stadia ratio of 1:100 is shown in fig. 10.

Fig. 10.— Solar Reticle

The solar circle has a radius of 15'-45". This is equal to the sun’s semidiameter when it is at a minimum approximately July 1.

The sun's image can be centered very accurately by superimposing the circle over the image of the sun. The Bureau of Land Management in the Manual of Instructions For The Survey of The Public Lands of The

United States 1947, described this advantage of the solar reticle. "The manipulation of the vertical and horizontal tangent-motions to the position of concentric fitting, of the circle to the sun's image may be accomplished with utmost certainty that the values for the vertical and horizontal angles are exactly simultaneous,"

The use of the solar reticle offers several other advantages.

Observations are faster and both horizontal and vertical angles -are read to the sun’s center. Unlike the quadrant-tangent method, there is no difficulty in selecting the correct quadrant or using a stadia line by mistake. Any single reading may be reduced separately without a correction for semidiameter. In the event of a suspected misreading of an angle, the difference between the several sightings, in travel timevertical angle, and horizontal angle, which should be propor tional, may be quickly checked.

Simplex Solar Shield

Professor C, H, Wall of Ohio State University developed a shield to be used for pointing at the center of the sun. It consists of a perforated shield which is mounted between the eyepiece of the transit and a solar screen. The perforations and other sighting points are so arranged that when a selected pair of these points are brought tangent to the sun's image, the center of the sun's image is

^Bureau of Land Management, Manual of Instructions For The Survey of The Public Lands of The Unite'd States ,"page 1301 28 on the horizontal or vertical crosshair» Davis and Foote9 in their well known surveying textbook, show a diagram of successive positions of the sm V s image with relation to the Simplex solar shield e ^

An effort was made to obtain a Simplex solar shield to compare the accuracy of pointing with other methods» Neither Carl Fti Purtz of the Civil Engineering Department g Ohio State University s or

Mrs 6 Co Ho Wall, wife of the late Professor Wall* could give any information concerning their manufacture or use®

Roelofs9 Solar Prism

One of the most recent and refined methods of pointing on the sun was introduced by Professor R 0 Roelofs of the Technical University of Delft 9 Holland* It consists of an attachment that fits over the objective lens of the telescope 6 The attachment contains a series of prisms that when pointed at the sun produces four images of the sun«

The overlapping images form a bright cross with a small dark square at the center. Fig, 11 shows this image as seen through a telescope*

^Davis and Foote * op.cit,* page 520 29

Fig, 11.— Reelofs’ Solar Prism

The Roelofs* prism offers several advantages over other methods of pointing on the sun.

1. The prism contains filters which produce monochromatic

green images of the sun which are comfortable to the observer's

eye and reduce the sunlight and heat which enter the telescope.

When using any other method of pointing a telescope at the sun,

the objective lens acts as a burning glass and causes extreme

heating of the reticle. This heating may result in an irregular

expansion of the crosshairs and an error will occur in both the

vertical and horizontal angles.

2. There is no confusion as to the choice of the quadrant 30

or correction necessary for s@midiameter«

3e The overlapping images provide a better target and result

in a more accurate pointing0

In Professor Roelofs9 book, Astronomy Applied to Land aaacraar ■ -iit c-Tirt i:. .rj sm m rwi&MJS n j=r. —iZaxriX zaJK1:. r Trrv - t^3gf?y,-tcu»«e;c«tt=»“»

g he states that a correction is necessary for the eccentri-

city of pointing when using the Roelofs’ prism.^ This correction is no longer necessary when using the prism as manufactured at the present times Wild Beerbrugg Instruments9 Inc, obtained the sole manufacture rights, and made a minor modificatione Correspondence with the Wild Beerbrugg Company resulted in the following explanatione

After Professor Roelofs turned over his prism idea to WILD for fabrication * we found there is an easy way to bypass, the eccentricity in pointing by inserting another wedge over the whole objective, That third wedge is tinted and serves as a sun filter at the same time. This wedge reflects the centre point of the four sun images back to the centre of the telescopec Actually in the WILD solar prism the centering wedge consists of two (one in front of the half wedges according to Roelofs and another behind if) for easier adjustment in the fabrication.

1 R. Roelofs s, Astronomy Applied to Land Surveyinga N» V, Wed. J, Ahrend 6 Z o o n 9 Amsterdam9 Holland, 1950, page 70. CHAPTER IV

CORRECTIONS FOR THE SUN'S

CURVATURE AND SEMIDIAMETER

Semidiameter

The sun's angular semidiameter is defined as the angle subtended at the earth's center by the sun's radius. Fig. 12 shows this rela tionship.

Earth Sun

Fig. 12.— Semidiameter Correction

The semidiameter (S) can be expressed as

r sin S = — P where

r = mean radius of the sun and

p = distance from the earth's center to the sun's center.

31 32

Since the path of the earth8s orbit is elliptical the distance p is constantly changing* This is reflected in the value of S* The range of S is between 158”45" and 168-17"e The exact value is given

in the solar ephemeris*

The ephemeris published by the Bureau of Land Management

tabulates the value of S ’at Greenwich apparent noon for each day of the year. The value is given to the nearest hundredth second*

The Keuffel and Esser solar ephemeris lists the semidiameter

for every ten days to the hundredth minute. The maximum change in any ten day period is three seconds.

The Gurley ephemeris gives the value of S to the nearest second for the first day of each month. The maximum change in semidiameter in any month is eight seconds, Straightline inter polation will give a value sufficiently accurate for all but the most precise survey work.

Correction to Vertical Angles

Vertical angles turned to sun9s upper or lower limb must be corrected for semidiameter» This correction is simply the tabulated semidiameter as taken from the ephemeris.

Correction to Horizontal Angles

The correction to a horizontal angle turned to the sun9s

limb is a function of the sun8s altitude. The relationship between the sun's semidiameter and altitude is shown in Fig, 13* 33

North South

Fig. 13,— Horizontal Angle Correction for Sun's Angular Semidiameter

The zenith distance (z) is by definition equal to 90°-h.

The law of sines applied to the right triangle formed by the center of the sun, the edge of the sun, and the zenith gives

sin A B _ sin 90° sin S sin z or

sin A B = sJ^Ll...... (9) sin z

Eq. (9) can be simplified by the assumption that the sin of a small angle is equal to the angle expressed in radians. Since the con 34

version factor for radians will cancel9 the correction to horizontal angles for semidiameter expressed in the -same unit as S is equal to

B ~ t S x esc z

B •** g,,, S X SeC h eoeeeeseeoeeodeeoeeoeosoeGeeAooeeeeeecee C 10 )

Curvature Correction

The authors of most surveying texts dealing with astronomical measurements recognize the error in assuming the path of the sun is a straight lineo They frequently place some time limitation on how

long a series of observations may extend when using the mean hori

zontal and vertical angles of such a series• There is little agree

ment as to the length of time the sun*s path may be assumed straight without introducing significant error* A time limitation of ten

minutes is probably the most frequently used, ^*2*3

The effect of using the average horizontal and vertical angles

of a series of observations is shown in Fig, 14, For simplicity, only

two positions'of the sun are shown,

^Bureau of Land Management § Manual of•Instructions for the Survey of The Public Lands of The United Statesg 19479 U, S 6 Government Printing Office $> page 528,

^Charles B, Breeds Surveying& John Wiley and Sons* Inc* 9 New York * 19429 page 141*

^Geologic Survey9 Topographic Instructions Solar Observations for Transit Traverse» Government Printing Office % 1953* page 9 & 3S

Horizon

Fig. 14,-- Effect of Curvature

Point (a) represents the average horizontal pointing on the sun. The corresponding average altitude (h) is then used to compute the sun's azimuth. This results in a computed azimuth (B) to point

(b). The difference (C) is known as the curvature error.

Paul Hartman, in a recent paper published by the American

Society of Civil Engineers, investigated the magnitude of this error and derived a formula for curvature correction.^ The equation is

^Paul Hartman, "Solar-Altitude Azimuth", Journal of the Surveying and Mapping Division, ASCE, Vol. 89, No. SU1, Proc, Paper 3410, February, 1963. This formidable appearing equation is solved by parts using a slide rule and tables. The first expression

A - la-n-P.H— x (1 ♦ COS2 B) dh2 sin B

2 2 2 (tan Phi. + tan h) _ sec h (12) tan B x sin B tan B and the second term

dh _ cos Phi, x sin t x cos Dec...... (13) dt cos h are solved by slide rule. The last term

At (14) i l is computed by use of a table. The most complete table is found in

Special Publication 14, of the U. S. Coast and Geodetic Survey.

When (B) is the azimuth of the sun measured from the north, east in the A.M. and west in the P.M. , the curvature correction (C) should always be added tc the value of (B) computed from the mean altitude (h).

Semidiameter Correction

The semidiameter correction applies only to solar observations where the quadrant-tangent method of pointing is used. Fig. 15 shows the effect of using the average of two horizontal angles turned to the sun’s limb.

Point (a) represents the bisector of the sun’s centers. Point 37

(b) is the bisector of the sun's limbs. The two points do not coincide

since S x secant h^^zf S x secant h^. This error is known as semi

diameter error.

•H rV

Horizon M

Fig, 15.— Effect of Semidiameter

Paul Hartman also derived a correction factor to be used for semidiameter.^ This factor (Cr) is obtained by solving the following equation.

Cr = ~ [0.127 x h x 105 + 0.322 x h3 x 109 +

0.60 x h 5 x 1013 + 0.99 x h 7 x 1017] ^ | A t | (15)

^Ibid., page 12. 38

In this equation

S = sun’s semidiameter in seconds

n = number of telescope pointings

h ~ sun’s altitude in degrees

t = difference of individual times of pointing from the

mean time*

The expression dh/dt is given in Eq» (13)*

Hartman states that this equation should be solved by slide rule® For values of (h) less than 30° only the first two terms inside the brackets are used* When (h) is less than 40° the first three terms are used*

The correction (Cr) should be added to the mean clockwise angle for an A.M* observation and subtracted for a P.M, observation if the west limb of the sun is used for the initial sighting*

Application

When using the quadrant"tangent method of pointing the effect of curvature error and semidiameter error tend to cancel one another provided the observing quadrants are properly chosen» Fig* 16 shows the sun as viewed directly through an erecting telescope in the correct quadrants *

In the northern hemisphere the first telescope position of a pair of pointings is that one which requires the sighting of the west limb of the sun * By observing9 in this manners the horizontal and vertical angles in the direct position, of the telescope will have approximately the same values as in the reversed position of the 39

telescope. This results from the movement of the sun during the time taken to plunge the telescope and make a second sighting. The semi diameter error and curvature error are a minimum when the corresponding angles in the direct and reversed positions of the telescope are about

the same.

A.M. P.M.

Fig. 16.— Quadrant-Tangent

The above method of pointing differs from that shown in

Fig. 8. Pointing as illustrated in Fig. 16 involves tracking the

sun with both the horizontal and vertical motions of the transit.

The curvature and semidiameter error are kept at a minimum only by

sacrificing the accuracy of pointing.

Curvature and semidiamcter corrections must be made if

accurate azimuths are to be determined. If high accuracy is desired

the altitude method should not be used. When the altitude method is

used there are several ways to overcome the effect of both curvature and semidiameter error0

The semidiameter.error will.not exist if pointings are made ■ on the sun9s center* This can be accomplished by a number of devices as pointed out in Chapter III* If pointings are made on the sun ?s limb then each pointing should be corrected to the smVs center*

This not only eliminates semidiameter error, but also provides a means of spotting a misreading of an angle or a poor pointing*

Curvature correction is unnecessary if each observation is used independently to compute azimuth*

When using the quadrant-tangent method of pointing the average of each pair of observations taken in diagonally opposed quadrants should be used to compute azimuth * The time duration will be so short and the vertical angles so nearly equal that any correction can be safely neglected*

Consistent azimuth determination by use of the altitude method demands a low dB/dh value* The corrections for curvature and semi diameter are relatively unimportant when the dB/dh ratio is one or less *

One apparent advantage of the curvature correction is in building up the horizontal angles * An engineer’s transit reading only to one minute could be used to measure the horizontal angle to the sun’s center within 10 or 15 seconds if sufficient repetitions were made * The curvature correction could then be applied to the azimuth computed by using the- mean altitude (h)0 This apparent advantage is lost when it is realized that the computed azimuth is dependent on the mean altitude» Regardless of the number of repetitions the vertical angle to the sun9s centers using a one minute transit % will be no better than one minute

^Winfield H e Eldridge§ "Discussion of Solar-Altitude Azimuth" Proceedings of the Surveying and Mapping Division, No* 3410§ ASCE§ Oct., 1963% ^ CHAPTER V

DETERMINING DECLINATION, LATITUDE,

AND LONGITUDE

Declination

Declination has already been defined as the astronomical position of the sun north or south of the celestial equator* Dec lination is independent of the observer's position as, at any particular instant, it is the same to all observers in all parts of the world*

The values of the sun9s declination for any given day are published in the ephemeris* The word ephemeris is of greek origin meaning diary or calendar* It contains the computed astronomical positions of Polaris, the sun s and a number of major stars for every day of the year.

The ephemerides published each year by the surveying instru ment manufacturers, such as Keuffel £ Esser and W. £ L. E. Gurley, give the sun's declination for 0 hour Greenwich Civil Time. This is the most convenient form for the present day surveyor who determines time by a radio time signal* Stations such as WWV give the civil time, and knowing the time zone, the Greenwich Civil Time can be easily found.

Before the event of wide usage of the radio, it was customary

* 42 43

to determine time by an altitude observation of the sun* The time so

determined was Apparent Time* In this case, it was more convenient

if the declination was given for Greenwich Apparent Noon or 0 hour

Greenwich Apparent Time* This is also the case when a solar

attachment is used and the hour angle must be set off in local

Apparent Time, Since the Bureau of Land Management uses the telescopic solar transit extensively in its work* the ephemeris published by them

gives the sun9s declination for Greenwich Apparent Noon,^

Declination as listed in an ephemeris is called apparent

declination* This means the declination of the sun is measured from

the true celestial equator* Apparent coordinates include all the

effects of proper motion* luni-solar and planetary precession*

nutation* and aberration* It is the apparent coordinates of the sun

that a land-surveyor must use in the astronomical determination of

latitude* longitude* and azimuth.

In all American ephemerides it is the practice to give the

change in declination per hour, Plate 1 is a sample form used in the

computation of declination. It is to be used when the sun 93 dec

lination is listed for 0 hour Greenwich Civil Time and the rate of

change in declination per hour is given. The use of such a form saves

time and prevents errors for those surveyors who take solar shots

infrequently,

^United States Department of the Interior* Bureau of Land Management* Ephemeris of The Sun* Polaris* and Other Selected Stars* United States Governme'nt Printing 0 ffice» 44 Plate 1

COMPUTATION OF SUN’S DECLINATION

Watch time of observation hr. _sec.

Watch error**- .,oeeoeeeeooeeoooeieoeooeeeoeQpeBoeQ min. sec.

Standard time of observation o e o o o o o o e o Jir._ m m , sec. 2 Longitude of central meridian oefroeyffpoeeoeeoi _hr.

Total time since 0 Greenwich e e o e c e o B e Jir«= m m , sec.

Total hours since 0 Greenwich hr.

Apparent Dec. for 0 Greenwich =« ©eeeeeoooeoeo

TfotaX hours) (Change in Dec./HrT)^

S lin ^ S DG C I m a t l on eeeeeeeoeoBeeoeeooeeo

^Watch error by comparison with radio time signal* Add difference if watch is slow and subtract difference if watch is fast<

^Longitude of central meridian expressed in time west of Greenwichi

Eastern Standard Time „,..»,„. 75 meridian *.* *« 5 hre Central Standard Time «0 ©*«>©© s 90 meridian Q Q»»© 6 hr e Mountain Standard Time ...... 105 meridian ..... 7 hr. Pacific Standard Time ...... 120 meridian ..... 8 hr 0

^Sign as given in Ephemeris, When using a solar attachment * the sun9s declination corrected for refraction in polar distance9 is usually computed or plotted in advance* The value of the sun*s refraction is added algebraically to the tabular declination. Therefore9 it will increase north or plus declination and decrease south or minus declinations,

. Plate 2 is an example of one method of plotting a declination curve to be used with the solar attachment* The straight line is the declination as taken from an ephemeris with a slope equal to the rate of change in declination per hour* The curved line is the declination corrected for refractiono The corrections for refraction are a function of the altitude of the sun* They can be taken directly from tables using an argument of declination* latitude* and hour angle*^

When an observation is to be taken* the time is observed and the corrected declination taken directly from the graph.

The effect of errors in declination on the computed azimuth will be covered in Chapter V I e

Latitude

Latitude is the angle between the direction of the plumb line and the plane of the earth*s equator* Latitude is positive when

^United States Department of The Interior* Bureau of Land Management* Standard Field Tables, U, S, Government Printing Office, Washington 25 * D, C, r T* — j— —| — I : T 4 — ^ I ait e __2 i' ~i i

4__ Te escopic . So I dr becUnatlion ; Setting 46

March,a; I 9 e 4 Ph i,= 3 2° -15

, S 6 ° ’5 0

US-7 i OO

M. Sv T. 47 measured north and negative when measured south of the equator.

In the United States, latitude can be scaled from a U.S.G.S.

7 1/2 minute quadrangle of 1:24,000 scale with an uncertainty of not more than * one second. This is based on the following reasoning:

Standard map accuracy for horizontal control requires that the map position of 90% of checked points be within 1/40 of an inch and that the other 10% be within 1/20 of an inch of their true positions.

Referring to Fig.17 and using the mean radius of the earth as

3,959 miles the coverage of a 7 1/2 minute quadrangle can be found.

Pcle

Phi Equator 3,959 mi

Fig. 17.— Accuracy of Measuring Latitude 4 8

2 x 3.14 x R - X or X = 8.637 miles 360 x 60 7.5'

Therefore, at a scale of 1:24,000, the length of a 7 1/2' quadrangle would be:

■?.16 .32. *. z 22.80 inches 24 ,000

This would mean that any position that can be located within 1/20 of an inch on the map represents the following length of arc.

22.80 in. _ 1/20 in. 7.5 x 60 X

X = 0.99 seconds

On a 15 minute quadrangle with a scale of 1:62,500 the above reasoning results in a maximum error of t 2,6 seconds.

The effect of an error in latitude on the computed azimuth will be considered in Chapter VI.

Longitude

Longitude can be defined as the angular distance measured along the equator from a fixed meridian to the meridian of the obser ver. The fixed meridian is usually considered as that meridian through

Greenwich, England, and longitude is considered positive when reckoned westward from that point.

The accuracy of determining the longitude of an instrument station by scaling from a map is dependent on the latitude of that station. For a given scale map the closer the station is to the equator the more accurate longitude can be determined. 49

Excluding Alaska the maximum error in longitude in the United

States , obtained by scaling from a U.S.G.S. map, would occur at a

latitude of 49°, The magnitude of this error can be computed by

referring to Fig. 18,

Pole

Phi Equator

Fig. 18.— Accuracy of Measuring Longitude

The circumference (Cir.) of the earth at a latitude of 49° equals

2 x 3.14 x r

Using the mean radius of the earth (R) as 3,959 miles and substituting

for r

Cir. = 2 x 3.14 x R x cos 49° = 16,312.3 miles

The coverage of a 7 1/2 minute quadrangle (X) would be 50

X _ 16,312.3 7.5 360 x 60

X = 5.66 miles

At a scale of 1:24,000 the length of this quadrangle would be

= 14.9 inches 24,000

If a position can be located on the quadrangle to an accuracy of 1/20

of an inch then the resulting error in longitude (L) would be

L _ 7.5 x 60 .05 “ 14.9

L = 1.51 seconds

The same reasoning will result in a maximum error of 1.17 seconds at

Tucson, using a latitude of 32°-151.

If the procedure is repeated for a 151 minute quadrangle at a

scale of 1: 62,500 the following results are obtained.

Latitude Error in Longitude from Scaling

32°-15’ ------3.08 seconds

49°-- ---————————— 3.95 seconds

When determining azimuth by the altitude method there is no

need to measure longitude. On the other hand use of the hour angle

method requires the accurate measurement of longitude. The time of

the observation together with the longitude of the station are used to

compute the local hour angle. The effect of an error in hour angle

on the bearing of the sun will be considered in Chapter VII, CHAPTER VI

AZIMUTH BY THE ALTITUDE OF THE SUN

Introduction

The determination of azimuth by the altitude of the sun is the most commonly used method in the United States. Unlike the hour angle method it does not require an accurate measurement of time. Survey ing textbooks and solar ephemerides published annually by instrument manufacturers explain this method in detail. The civil engineering student may hear of other methods of determining azimuth by solar observation, but it is highly probable that in his college course work, the altitude method is the only one he will use.

Trigonometric Formulas

The basic equation for the solution of the astronomical tri angle when the altitude is known was derived in Chapter II. It will be repeated here along with a number of other equations used in the altitude method. The equations, though different in appearance, are basically the same. All equations require knowing the latitude of the station and the altitude and declination of the sun. 52

c o s B s ..... A ^ .n Dec* ^ .. tan h x tan Phi...... (17) cos h x cos Phi.

tan 2 1/2 B = sin(s-h) x sin(s-Phi.)...... (18) cos s x cos(s-p)

sec 2 1/2 B = (19) sec h x sec Phi.

vers B = sec Phi. x sec h [vers p - vers(Phi.-h)] ..... (20)

Each term in Eq. (5) has already been defined. The value of

(h) used in this equation and all equations involving the sun’s alti

tude must be corrected for refraction and parallax. In Eq. (5)

azimuth is measured from the north. If the observation was taken in

the morning the bearing is east of north, if taken in the afternoon west of north. When a minus value results from the solution of Eq. (5)

azimuth is measured from the south, east in the A.M., and west in the

P.M.

Eq. (16) is identical with Eq. (5) with the exception of the

sign. In this case a positive sign indicates the azimuth is measured

from the south, and a negative sign means the azimuth is measured from

the north. Again it is measured east in the A.M. and west in the P.M.

Dividing each term in the numerator of Eq, (5) by cos h x

cos Phi. results in Eq. (17).

An interesting variation of Eq. (17) was derived by Philip

Inch.^ Starting with

cos B = Sj-P. --- tan h x tan Phi...... (17) cos h x cos Phi.

^Tnch, op.cit., page 971 The resulting equation was

cos B = A x sin Dec. - B ...... (21)

Tables were then arranged using arguments of (h) and (Phi.).

Values of (h) from 15° to 55° were plotted against values of (Phi.) from 31° to 49°. Values of (A) and (B) could be taken directly from the tables. The tables included correction for refraction and parallax.

Eq. (18) is an application of the half angle formula to Eq.(5).

A step by step derivation can be found in most texts of spherical trigonometry.* The polar distance (p) = 90° - Dec. and s = l/2(p + h

+ Phi.) are used for the first time in this equation.

Eq. (19) is similar to Eq. (18) but involves only one trig onometric function, the secant. It was derived by T. F. Nickerson by a number of substitutions in the basic half angle formula for cos B/2. Eq. (19) has two advantages over any of the other altitude

^William L. Hart, College Trigonometry, D. C. Heath and Company, Boston, Mass., 1951, page 193.

^T. F. Nickerson, Determination of Position and Azimuth by Simple and Accurate Methods, Transactions of A&CE, Vol. 114, 1949, page 143. 54

equations« It deals entirely with one function and in case logarithms

are used the characteristics will always be positive. Plate 3 is a

sample form to be used in computing azimuth by Eq. (19). The form is

patterned after that of Hickerson in his book* Latitude* Longitude* and

Azimuth by The Sun or StarsA table of log secants is used and the

solution requires no multiplication. To those who use logarithms

- infrequently the use of such a form, saves both time and costly errors,

Eq, (20) makes use of the versed sine (1 minus the cosine). By

exchanging the polar distance (p) for 90°- Dec, and use of the double

angle formula it reduces to Eq. (5).

Factors Affecting The Measuring of Altitude

The altitude method of solar observation is directly dependent

on how good the vertical angle can be measured to the sun’s center.

There are a number of factors that can contribute to incorrect vertical

angles. These factors will be considered"in the following paragraphs, -

Instrument Error

The first source of error would be in the adjustment of the

instrument. There are three major conditions that result in erroneous

vertical angles.

1. The vertical axis out of plumb.

^T. F. Hickerson„ Latitude* Longitude* and Azimuth by The Sun

or Stars * published by the Authora Chapel Hill* N.C, 9 1947* page 40, "O < 'O Add r- h. = Phi. Dec. — = S—h = S-P l. Sum Alg. 8 = 28 = h = S = P Nrh e. s ad s subtracted. is and + is Dec. ^North = oiotl nl fo ot rcoe es i AM and A.M. in east reckoned north from angle Horizontal = B = /( + + Phi.) + h + 1/2(P = S = 90o-Dec. = P ot Dc i - n i added. is and - is Dec. South

et n P.M. in west Sec. 9- 9- 60" 59'- 89°- 2 AZIMUTH OF DETERMINATION By Use of Log Secants Log of Use By / B = B l/2 lt 3 Plate / = Z 1/2 Subtract = Z Pi (check) Phi. =

Subtract

o e. 10' x Sec. Log

q 1 55 i < XI T3 56

2» A lack of parallelism between the line of sight and the axis of the telescope level,

3<> The displacement of the vertical vernier from its adjusted position.

The first condition,.inclination of the vertical axis, is the most serious. Unlike the other two sources of instrumental error$ it cannot be eliminated by observational procedure. It can be reduced to a negligible amount by careful leveling of the transit.

The most exact method of leveling a transit is by use of the telescope bubble. The procedure is as follows: After the instrument is leveled using the plate bubbles a the telescope is brought over a pair of leveling screws with the (A) vernier set to 0°, Center the telescope bubble by using the vertical clamp and tangent screw. The upper motion is released and the telescope rotated 180° as shown by the reading on the (A) vernier. If the telescope bubble is not centered» bring it halfway to the center by use of the vertical tangent screw and the remaining way by using the two leveling screws. The upper motion is released and the telescope rotated to its initial position.

The telescope bubble should remain centered. If it does not 8 repeat the process of bringing it halfway to the center by use of the tangent screw and the remaining way by use of the two leveling screws. When

the bubble remains centered at both 0 ° and 180° the telescope is rotated until the (A) vernier reads 90° or 270°, This places the telescope in line with the other pair of leveling screws. The 57 telescope bubble is now centered by use of the leveling screws only.

This completes the operation and makes the vertical axis of the transit truly vertical.

The error resulting from measuring a vertical angle when the transit has not been carefully leveled is shown in Fig. 19.

Angle (H) is the angle of inclination of the vertical axis.

The axis lies in a vertical plane that deviates from the vertical plane containing the sun by the angle (Z). The error in the measured

vertical angle is equal to (H) when (Z) = 0 ° and 0° when (Z) = 90°.

Sun

Fig. 19.— Instrument Error 58

The angle (H) can be accurately measured by means of the tele scope bubble« The sensitivity of the bubble is the angle of inclina tion in seconds of arc per division of bubble run« This means that when using an Engineer’s transit, with a telescope bubble having a sensitivity of 60 secondse the displacement of the bubble by one-half of a division could result in an error of 30 seconds in the vertical angle.

The second and third condition can be corrected by adjustment of the transit or eliminated by the observational procedure.

The adjustment for a lack of parallelism between the line of sight and the axis of the telescope level is known as the peg adjust ment. It is explained in detail in any surveying textbook or instru ment manual.

The Vertical vernier can be checked for adjustment after the instrument has been carefully leveled using the telescope bubble.

When the bubble is centered, the vertical angle should be 0 °, If it is not the vernier is loosened and moved until the correct reading is obtained.

The error resulting from a lack of parallelism between the line of sight and the axis of the telescope level, or from displace ment of the vertical-circle vernier, or a combination of the two,.is known as index error. Adjustment for index error is unnecessary if the transit has a full vertical circle. When the vertical axis of the transit is truly vertical, the mean of two vertical angles, one 59 taken with the telescope direct and the other with the telescope

reversed 9 is free from index error.

Another possible source of error would be imperfections in the manufacture of the instrument. Such things as eccentricity and imperfect graduations are relatively unimportant in a modern transit in good adjustment and can be ignored in all but the most precise work.

The magnitude of the instrument error is almost completely dependent on how carefully the transit has been leveled when using correct procedure in measuring a vertical angle.

Refraction

As light travels from the sun to the earth, it passes from the empty interstellar space into the earth’s atmosphere. As the height above the earth's surface diminishes„ the density and tempera ture of the air increases. This results in the ray of light being bent vertically downward. There are a number of equations that have been derived to correct for this phenomenon. The basic equations are based on a number of assumptions. It is assumed that the atmosphere is built up of an infinite number of infinitely thin spherical layers concentrical with the earth, and each having a uniform refractive index throughout.

An equation can be derived by ignoring the curvature of the layers and assuming the number and thickness to be finite. Applying

Snell's Law of refraction to Fig. 2 0 * the following equation can be written. 60

Earth’s Surface

Fig. 2 0 ,— Refraction

sin Z,, U 3 7 n r r 3 * un

In this equation (U) is the index of refraction and in the case of a vacuum is equal.to 1. Therefore,

t sin Zn = sin Z 3 x U 3 ...... (22)

If Snell's Law is applied to the succeeding layers the following equations can be written:

U 3 x sin Z 3 = U 2 x sin Zg

U 2 x sin Z 2 = Ujl x sin Z^ by substitution Eq. (22) becomes

sin zn ° x sin Z ^ ...... (23) Cl

since

refraction (r) = Zfi -

zn = zx + r

Eq. (23) becomes

sin (Z^ + r) = x sin Z^ ...... (24)

Expanding Eq. (24) by use of the addition formula results in

sin Z jl x cos r + cos Z^ x sin r = x sin Z^

Since (r) is a small angle, usually less than 35 minutes, the cosine

of (r) is approximately one. Therefore,

sin Z^ t cos Zjl x sin r = x sin Zj

If refraction (r) is expressed in radians then for a small angle sin r

is approximately equal to (r). Then

sin Z jl + cos Z jl x r = Ui x sin Z^

r = tan Zjj x (U^ - 1) ...... (25)

It should be noted that in this formula that only the index

of refraction for the lowest layer appears. This equation gives good

results when the zenith distances are so small that the assumption of

horizontal planes is reasonably correct.

An empirical equation has been derived by Comstock that gives a closer approximation.^

^Nassau, op.cit., page 66, 62

r - HH t x tan z / ...... (26) where

r = refraction in seconds of arc.

b = barometric pressure in inches,

t = temperature in degrees Fahrenheit.

= observed zenith distance.

According to Nassau, for zenith distances under 75°, Eq. (26) should give the refraction within one second.

A number of tables are available that list the mean refraction and give both temperature and pressure corrections. Two of the most complete tables are found in U.S.C. & G.S. Special Publication No. 247 i o and in Seven Place Logarithmic Tables by Von Vega. »

When solar observations are taken with a one minute transit the surveyor should not be overly concerned with the temperature and barometric pressure corrections to the tabulated refraction. Unless observations are made at extreme temperatures and at high elevations the corrections are relatively unimportant. Host of the azimuth determinations in the United States are made at an elevation between

sea level and 2000 ft. and temperatures between 0 ° and 100° Fahren heit. This would result in a maximum error of 18 seconds if correc-

^U. S. Coast and Geodetic Survey, Special Publication No. 247, Government Printing Office, Washington 25, D. C., Tables 25, 26, and 27.

^Baron Von Vega, Seven Place Logarithmic Tables, D. Van Nostrand Company, Inc., Princeton, New Jersey. tions were ignored for an altitude observation of 15°.

The correction for refraction is always subtracted from the

observed altitude.

One of the most common sources of error in computing refrac

tion is due to a local variation in temperature and pressure. This

variation can be caused by the close proximity of lakes, forests, or buiIdings.

Parallax

Parallax is the correction to the observed zenith distance

to the sun, measured from a point on the earth's surface, necessary

to represent the zenith distance from the earth's center. The

parallax of the sun for any observed zenith distance can be obtained by referring to Fig. 21.

Sun Horizon

Fig. 21.— Parallax 64

By the Law of Sines

sin p _ sin (180° - %/) " R “•------d...... (27) where

= observed zenith distance

R = radius of the earth

p = sun's parallax

d = distance of sun from the center of the earth

Again referring to Fig. 21 the horizontal parallax (p) is equal to:

sin p/ = £ ...... (28) d

From Eq, (27)

sin p = ~ x (sin 180° x cos - cos 180° x sin Z^) d or

R / sin p = — x sin Z' d substituting the value of R/d from Eq. (28) gives

sin p = sin p/ x sin Z ^ ...... (29)

The horizontal parallax of the sun is given in the American

Ephemeris for each day of the year. It has a maximum value of 8.95

seconds and a minimum value of 8 . 6 6 seconds. This gives an average value of 8.80 seconds. In place of computing the sun's parallax by

Eq. (29) a satisfactory expression can be obtained as follows. Using the average value of the sun's horizontal parallax and realizing that the value of (p) will always be less than p/, an approximate solution 65 to Eq. (29) can be written

p = 8.80 sin ...... (30)

The parallax correction is added to the vertical angle to correct the altitude observation to the center of the earth.

Altitude Correction for Semidiameter

If in making a solar observation the limb or edge of the sun is observed, the vertical angle must be corrected to the center of the sun. If the upper limb of the sun is used, the semidiameter is subtracted from the altitude. If the lower limb is observed, the semidiameter is added to the altitude.

The sun's semidiameter is dependent on the distance between the sun and the earth. This distance is constantly changing due to the ellipticity of the earth's orbit around the sun. The semidiameter is tabulated in the solar ephemeris for each day of the year.

Effect of Errors in Altitude on the Computed Azimuth

The equation most frequently used to compute the sun's azimuth by the altitude method was derived in Chapter II. This equation

n - sin Dec. - sin h x sin Phi. , c\ cos ti - .... — ■ ■ v” ...... I,— lb; cos h x cos Phi. can be differentiated with respect to (h) holding (Dec.) and (Phi.) constant. After simplifying the equation becomes

dB tan Phi. - tan h x cos B , dFT ------sTiTB...... (31) 66

As the sun's bearing approaches the meridian at noon the value of (B) will approach 90° and an error in altitude will have the greatest effect on the computed azimuths

Since the measured altitude to the sun’s center is subject to a number of factors that may introduce error, the dB/dh ratio is probably the most important criterion as to the accuracy of the altitude method. Inspection of the dB/dh ratio will tell the surveyor the possible error in a computed azimuth for a given probable error in the measured altitude, <

The dB/dh ratio can be obtained at the time of the observation by comparing the difference in two horizontal angles to the sun’s center with the difference between two corresponding vertical angles to the sun’s center, A more useful method would be one that would tell the surveyor at what time, on any given date, he must take an observation to obtain a desired dB/dh ratio. This can be accomplished by means of an altitude-azimuth curve.

Inspection of Eq, (31) shows that the dB/dh ratio is dependent on the latitude of the observing station. Sufficiently accurate altitude-azimuth curves can be developed by using the latitude of a central station in the area of interest.

The curves can be developed by either of two methods, Eq, (5) can be solved for (B) by assuming different values of (h) for a given declination and latitude. An easier method is by the use of prepared tables that give the values of (B) for a given (h), (Phi.), and

X (Dec.). If tables are used the hour angle is also given and may be plotted together with the altitude and azimuth. Eq. (31) is then solved for different values of dB/dh and the results plotted on the altitude-azimuth curves. Plate 4 is an example of the altitude- azimuth curves for Tucson, Arizona, with both hour angle and dB/dh curves superimposed.

Once such curves have been developed for a particular area, limits can be established on when observations should be taken. For example, a limiting value of the dB/dh ratio equal to 1.5 would be reasonable at Tucson, This would assure consistent results and at

no time require an altitude measurement less than 2 0 °»

Effect of Errors in Declination on the

Computed Azimuth < niri—B" m..., /T

When the altitude method of determining azimuth is employed, the only reason for observing the time is in the computation of dec lination, The maximum change in declination per hour is approximately, sixty seconds. This means that for the most accurate work, a deter mination of time within two minutes would result in an error no larger than two seconds in declination. When taking a solar shot with

•^Tables of Computed Altitude and Azimuth, Hydrographic Office Publicati^%or°°^T^T°W^T%a%yT^T?T°^^%Sent Printing Office, Washington, D. C. „ akljTUDE-AzlMUTh

Tucson, Arizona

rr-B 'm m g ^ d f'S u fT lfr6hii“ ]Squ™f>i r ' EijsTTn STM': v \W esT Tn"“R'Tff

I ; i : i . : • • • • ; I • ' ' I t • . ' ! • .. • I ... • . an engineer’s transit, with horizontal and vertical verniers reading to one minute, and the rate of change in the sun’s declination an average value of 25 seconds per hour, a determination of time within twenty minutes is normally sufficient. The effect of an error in declination on the computed azimuth can be determined for any latitude

Deferentiating the altitude Eq. (5) and holding (Phi.) and (h) constant results in the following expression.

dB _ cos Dec. (32) dDec sin B x cos k x cos Hii......

The above equation was solved for a latitude of 32°-15’ with

the aid of the altitude-azimuth curves. Plate 5 is a plot of dB/dDec. against the sun's altitude. Observations taken with a dB/dh ratio of two or less result in a maximum dB/dDec. ratio of 3.

By the use of Plate 5 the effect of an error in time can be quickly determined. Assume an observation was taken on October 10,

1963, with a dB/dh ratio of approximately 1. The sun’s declination for this date is - 60-17', The change in declination per hour is

56.93 seconds. The dB/dDec. ratio as taken from Plate 5 equals 1.5.

An error in time of 2 0 minutes would result in an error in declination of 19 seconds and an error in azimuth of 28 seconds.

When the altitude method of determining azimuth is being used the time can be obtained by calculation or by use of a nomograph such zmtt ot An: Err Th e S dBear in

a i T t r r f f

±L:_ j . r

Sun s A1111 u de 71 as that of Roelofs.^ The time obtained by use of the nomograph will be correct within two or three minutes if the longitude of the station is known,

Effect of Errors in Latitude on the

Computed Azimuth

The effect of an error in latitude on the computed azimuth can be found by taking the derivitive of the sun's bearing with respect to latitude. The following expression results from differentiating

Eq. (5) while holding altitude and declination constant,

dB _ sin h - sin Phi. x sin Dec. tqot ■ ■ ■» - — • * '■ ■ *#*##**»**########## \ VU J dPhi. sin B x cos^Phi. x cos h

A simpler method of expressing dB can be obtained by use of Eq, (4) dPhi. and Eq. (G).

cot t - cfs t sin h - sin Phi, x sin Dec. co ” sin t cos P'hi. x cos Tfi x sin B

By substitution

_dB = cot t = sec Phi...... ( 3 4 ) dPhi. cos Phi. tab t

Plate 6 shows the effect of latitude and hour angle on the dB/dPhi. ratio. It can be seen that at the sixth hour angle the dB/dPhi. ratio is always zero. For any other hour angle, the higher the latitude,

^Roelofs, op.cit., page 214. Effect of I fftnde-'tnri (IPhi.

4 9° -Ot

sec Phi tan t

— f *" - 1.'— i t % LU l JLU •— l - M - L L I - 4

Angle the greater the dB/dPhi« ratioe

Plate 7 is a plot of the altitude-azimuth curves for a latitude of 320~15# with the dB/dPhi, curves superimposed« Inspection

of the curves 8 shows that observations taken when the hour angle is greater than two, result in a dB/dPhi, ratio of two or less. When an uncertainty of latitude.exists, it is well to take solar obser vations either early in the morning or as late in the afternoon as possible.

Field Procedure for Observations

The instrument is set up over a selected point and carefully leveled. The horizontal circle is set to read 0°-00,-00". The lower motion is released and a sight is taken along the given line to a fixed reference point. The lower motion is locked, the upper motion released, and a pointing made on the sun. The sighting is accomplished by use of one of the methods in Chapter III. The hori zontal and vertical angle and the time are read and recorded. The instrument is again pointed at the fixed reference point and the horizontal reading checked to see if it agrees with the initial reading of 0o”00e-00,,o This would complete a single altitude

observation on the sun. Assuming the latitude of the station is known all necessary measurements have been made to determine the bearing of the given line. ■ r '• 4liLtu d e: ii: A z imjut fi

4iPhk Pilate

5 0 It should be pointed out that in actual practice a series of observations are made on the suns rather than relying on a single observation. The procedure for taking a series of observations varies widely and was discussed in Chapter III, CHAPTER VII

AZIMUTH BY THE HOUR ANGLE

OF THE SUN

Introduction

The hour angle method of determining azimuth by the sun has been rejected in favor of the altitude method by a large percentage of surveyors. This was understandable in the past when the only accurate method of determining time was by observing the meridian passage of the sun@ Polariss or one of the equatorial stars. The present day surveyor has little justification in overlooking a method that is superior to the altitude method in that it is limited in accuracy only by the pointing ability of the instrument used.

Determining azimuth by the hour angle method requires knowing the precise time of observation and the longitude of the observing station. The altitude of the sun, the factor which is subject to a number-of errors, is not used in computing azimuth by the hour, angle method.

Trigonometric Formulas

The basic equation was derived in Chapter II. It will be repeated here along with a number of variations. To conform to north as zero azimuth Eq. (8 ) should be rewritten

u _ sin Phi. x cos t - cos Phi. x tan Dec. / \ C O t D — 1 m mm mmmm u ■ ■ wi ■ ■■ i # # # # # # # # # # \ VU / sin t

If the solution of this equation results in a positive (B) then the azimuth of the sun is reckoned clockwise from the north.

Since the cot function is positive from 0° to 90° and 180° to 270° the sun will be in the first quadrant for A.M. observations and in the third quadrant for P.M. observations.

If the value of (B) is negative then the azimuth of the sun is reckoned counter clockwise from the north. This will place the sun in the fourth quadrant for P.M. observations and in the second quadrant

for A.M. observations.

Confusion as to the correct bearing of a line when using

Eq. (35) can be avoided by use of the following equations.

cot B is positive and A.M. observation

. Az. = B +.H...... (36)

cot B is positive and P.M. observation

Az. = B + H + 180° . (37)

cot B is negative and A.M. observation

Az. = B + H + 180° ...... (38)

cot B is negative and P.M. observation

Az. = B + H + 360° ...... (39)

The above equations will give the azimuth (Az.) of the line

measured in a clockwise direction from the north. The horizontal angle 78

(H) used in Eq. (36) to (39) must be the clockwise angle from the sun to the target. In Eq. (38) and (39) the value of (B) will be negative.

If Eq. (35) is inverted and both the numerator and denominator are divided by cos Phi. x tan Dec. then

tan B • sin t x sec Phi, x cot Dec...... (40) tan Phi. x cos t x cot Dec. - 1

By letting

tan Phi. x cos t x cot Dec. = a

Eq. (40) becomes

tan B « sin t x sec Phi. x cot Dec. x __1_ ...... (41) 1 -a

The solution to Eq. (41) is facilitated by use of tables

prepared for finding the log 1 / 1 -a directly from log a.

Determining the Hour Angle of the Sun

The local hour angle of the sun is the angle at the pole from the meridian westward to the hour circle through the body. Measurement of the sun's hour angle requires the accurate knowledge of both time and longitude. Since a number of kinds of time are in common usage, the surveyor must have an understanding of the measurement of time.

The measurement of time is based on the rotation of the earth on its axis. This rotation can be measured with respect to a number of celestial objects. If the sun is used to observe the earth's

lu. S. Coast and Geodetic Survey, Special Publication 14, U. S. Government Printing Office, Washington, D. C. 79 rotation the time is called solar time. Solar time is divided into three classesi apparent time* mean timeg and standard time.

Apparent time is based on the real sun. An apparent solar day is the interval of time between two successive transits, either

upper or lower 8 of the sun over the meridian of that place. Owing to the obliquity of the ecliptic and the lack of uniformity of the motion of the earth in its orbit the apparent time rate is irregular®

Mean solar time is based upon a fictitious or imaginary sun whose solar day is mathematically uniform. Since mean solar time is uniform and regular in its passage e clocks and watches in ordinary use are designed to be rated for a 24 hour period that conforms to the mean sun. Civil time has the same-meaning as mean solar time.

The equation of time is the difference in hour angle between the true sun and the mean sun. It is counted in the mean time rate.

The equation of time is changing constantly and its value for each day„ on the Greenwich meridian, is tabulated in the ephemeris.

Standard time is the mean solar time on the central meridian

of each time belt. Standard meridians, beginning at Greenwich,

England 6 are established each 15° of longitude around the globe.

Local mean time is identical with mean solar time on the meridian at the station where the time is being employed. Stations

that are 1 ° apart in longitude differ by four minutes in local mean

time.

If the time of observation is measured on a watch keeping standard time the following steps are necessary to determine the local hour angle.

The Greenwich civil time (G.C.T.) of the observation is found by correcting for the longitude of the central meridian, if the

observing station is in the United States then

G.C.T, = Standard Time + Longitude of Central Meridian .. (42)

The correction for the longitude of the central meridian must be expressed in time. Since the central meridians for the time zones are 15° apart the correction will be in even hours.

The equation of time for the day of observation is taken from the ephemeris and corrected for the time of observation. In both the

Gurley and K. 6 E. solar ephemerides the equation of time is given for midnight G.C.T, along with the correction per hour. The corrected value can be found by using the G.C.T, computed in Eq» (42)6

Eq, of Time 3 Tabulated Value t Diff./Hr, x G.C.T, (43)

The Greenwich apparent time (G.A.T.) is computed by use of the corrected equation of time, -

G.A.T. 3 G.C.T, t Eq. of Time ...e.,,,,,,,...,...,,,,.,,. (44) and

G.H.A. = G.A.T. - 12hr

The Greenwich hour angle (G.H.A.) is then used in calculating the local hour angle (t). The relationship between the two is shown in Fig. 22. 81

For longitudes west of Greenwich it can be seen that

t = G.H.A, - Longitude of Station ...... (45)

It is the local hour angle (t) that is used in computing azimuth by the hour angle of the sun.

Sun Local Meridian

Greenwich Meridian

Fig, 22.-- Relationship Between the Greenwich

Hour Angle and Local Hour Angle

Factors Affecting the Measurement of

the Sun's Hour Angle

As was seen in the preceding section the local hour angle is

computed by knowing the precise time of observation, the equation of

time, and the longitude of the observing station. Since an error in

any of these factors will result in an error in hour angle the

measurement of each factor will be considered. 82

Time of Observation

Several methods are available to the surveyor to determine the precise time of each observation. The method and equipment used in determining time should be consistent with the type of instrument and the desired accuracy of the azimuth determination.

In the most precise, work where an instrument such as the Wild

Tg is being used 9 the time of each observation is frequently measured by means of a chronograph. This is an instrument„ which by means of a

recording pen 9 produces a graphic record of the passage of time. Each

time an observation is made the observer presses a signal key which

causes an offset in the record which can be read to the nearest hund™ redth of a second. The chronograph is an expensive instrument and

its use is neither justified or necessary in most survey work.

The development of light, weight and moderately priced tran

sistor radios with short wave bands offer another method of accurately,

determining time. "It has been found that when using a simple stop watch and short wave receiver in the field, the time of sun pointings

can be determined to the nearest 0 . 1 second,"^

Time can be consistently measured within 0,2 seconds by use of

a stop watch and an accurate time piece. In the past it was necessary

to use a special built chronometer to insure accurate time keeping.

^Winfield H, Eldridge» "Purpose and Procedures for Meridian Determinations", Proceedings^ Illinois Land Surveyors Conf., RLSA, Vol. Ill, Urbana,™rTTT%^T^D2%™page™l5Tr^^™^™^°™™^ ™™^^^*™^™' 83

The chronometer was wound at regular intervals and the chronometer rate was to some extent dependent on its position. The recent development by the Bulova Watch Company of the Accutron wrist timepiece has made the need for a chronometer unnecessary. This watch operated by an electronically driven tuning fork has an amazingly uniform rate that is independent of position.

The least accurate method but the one most frequently used is time by calling out. The observer calls "time" at the instant the sighting is made and the recorder immediately reads the survey time

piece. Care must be taken to read the second hand 9 the minute hand* and the hour hand in that order. The accuracy of this method depends on the reflexes of both the observer and the recorder. Accuracy no better than the nearest second should be expected.

The Equation of Time

The equation of time is tabulated in the ephemeris for every day of the year. The ephemeris published by the Bureau of Land

Management simply list® the equation of time for Greenwich apparent noon for each day of the year. The ephemerides published both by the

Gurley and the Keuffel 6 Esser Instrument Companies list the equation

of time for 0 hour Greenwich Civil time for each date. They also give the hourly change in the equation of time.

. If an accuracy of 0.1 second is sufficient straightline interpolation can be used to find the equation of time for the time of 84 observation. If greater accuracy is needed a method of second differ ences must be used. Roelofs presented a nomogram to facilitate finding this second term.^

Longitude of Observing Station

The solution of Eq. (45) for the local hour angle involved the longitude of the observing station. It was seen in Chapter V that the maximum error in longitude, resulting from scaling from a

U.S.G.S. map, would be less than four seconds. It must be remembered that this four seconds is in terms of arc and not time. The equiva lent unit of time would be 0.267 seconds.

Probable Error

The probable error in determining the hour angle of the sun can be computed by using the above limits. If time is measured by means of a stop watch and an accurate timepiece, the equation of time is obtained by straightline interpolation, and longitude is scaled from a U.S.G.S. map the following probable error could be expected.

^ [(0.2)2 + (0.1)2 + (0.27)2]= ♦ Q.35"

Effect of an Error in Time on the Computed Azimuth

The effect of an error in time on the computed azimuth can be

found by differentiating Eq. ( 8 ) with respect to t, while holding

^Roelofs, op.clt., page 211. 85 declination and latitude constant. This gives the following equation.

dB _ cot t x cot B -f sin Phi. ^ ...... ^ (46) dt csc^ B

This equation was solved with the aid of the altitude-azimuth curves for a latitude of 32°-15f North. The results are shown in

Plate 8 . Inspection of the curves indicate that observations should be limited to an hour angle greater than two. This would give a maximum dB/dt ratio of approximately 1. At this ratio, a one second error in time will cause a 15 second error in azimuth.

If observations are taken only at an hour angle greater than four the maximum error in azimuth due to a one second error in time

would be 1 0 seconds. It must be remembered that the error in time could be in any of the three factors; time of observation, equation of time, or longitude of observing station.

Field Procedure for Observations

Since the hour angle method requires very accurate time control, it is essential that the survey timepiece be checked for accuracy both before and after an azimuth determination. The easiest method of obtaining the correct time is by means of a radio time signal.

A number of stations broadcast time signals in the United

States. The best known of these is probably WWV transmitting on frequencies of 2.5, 5, 10, 15, 20, and 25 megacycles from Rate

zt math Seconc Err or r '• * ‘ * + t ; * ; r ' ! '* H i t T H

I: III L a 4ti 111 jlB i 0 0 I — — L • ^ iij r.i± l T : n " r n . J—h+- r-4- •- - r r i t n ; # I 1" ; .r i- r--I- - t ] ; : m > 2 3 ’x +- <- 1 --j r t.tli: i.t' |:r l l r l l l l ‘ I ! t— (-- - I— t V I ■ 1 1 1 4 - ! 0 , "I : iltil: I

♦ - ♦ • 4 - j 4 - + i 4- i - --1 i- ♦ - i —j Jit’’.' : 1 -j ♦ -4- - I— |— +- *- — |— ♦- I-. |~— t 1 -* ♦ —* j- 1 t-'it-t-tt:;: T—[ -)- * "h ■* ♦- - |

I ■ M 1 : : f • « — 4 r-f- • • ♦ 4 n ■ Lili h'-B

- - 4 4- 87'

Washington, D„ C, They give the Eastern Standard Time every five minutes. Two new radio stations operated by the National Bureau of

Standards are now in operation. They are WWVB„ a low-frequency station transmitting at 60 kilocycles, and WWVL, the very-low-

frequency station, at 2 0 kilocycles. These stations, near Fort

Collins, Colorado, are now transmitting frequency signals accurate to a millionth, of a second, A system of equally accurate time signals will be added in the near future.

Checking the timepiece before and after a series of obser vations establishes the rate of the timepiece. The watch or chron ometer used may run fast or slow without harm as long as the time rate is constant. Straightline interpolation will give the correction to apply to the time of each observation.

Since the longitude of the observing station is used in computing the hour angle of the sun, a station should be selected that can be found on a U.S.G.S, map. The observing station can be an identifiable point on the map or it can be carefully tied to such a point. Either method will allow scaling the longitude.

The instrument is set up over a selected point and cafefully

leveled. The horizontal circle is set to read 0 °- 0 0 ,- 0 0 ,’» The Ipwer motion is released and a sight is taken along the given line to a fixed reference point. The lower motion is locked and the upper motion released* and a pointing made on the sun. Sighting on the sun is accomplished by one of the methods discussed in Chapter III, 88

If the observation is made by using a solar screene without the aid

of any sighting device 3 the center-tangent method should be used in preference to the quadrant-tangent method.

The observing procedure given will assume the use of the center-tangent method and a stop watch, The telescope is pointed somewhat in advance of the sun’s path. The observer holds the stop watch in hand and watches the sun approach the vertical wire. The moment the sun’s limb touches the wire the stop watch is started.

The stop watch is then compared with the survey chronometer and stopped on an even 10 second chronometer reading. The chronometer reading minus the stop watch reading is the time of observation. The horizontal angle is read and recorded. The vertical angle is read on the first and last observations of the series. The altitude is used for the semidiameter correction and can be computed from the sun’s hour angle, but in most cases it is easier to observe. The instrument is again pointed at the fixed reference point and the horizontal reading checked to see if it agrees with the initial reading of 0o-00’-00,,« This would complete a single observation on the sun by the hour angle method. CHAPTER VIII

OTHER METHODS OF DETERMINING

AZIMUTH BY THE SUN

Azimuth by the Altitude and

Hour Angle of the Sun

The simplest method of computing the azimuth of the sun is by use of Eq« (4)« This equation e D „ COS Dec. x sin t fat @ oeoeoesoeeoeoooeeeeooeoeoooeoo© \ ^ / cos h

involves only three quantities 5 the declination$ hour angle» and altitude of the sun. The latitude of the observing station is not required.

The method$ though simple in appearances has little practical value. The two hardest quantities to measure in the field, altitude and time are required. Since an error in the measurement of either of these quantities will affect the computed azimuth the accuracy of this method is rather limited. Computation of the sun's hour angle requires an accurate measurement of longitude. Since it is highly unlikely that the longitude of the observing station would.be known and not the latitude the one advantage of using Eq. (4) is lost.

Inspection of the dB/dh curves in Plate 4 and the dB/dt

curves in Plate 8 for a given hour angle and declination will give the

89 90 accuracy that can be expected from any observation,.

If the time of observation is not known within a few seconds the use of Eq. (4) will give absurd results. Any equation involving the sun’s hour angle requires knowing time within five seconds to keep the resulting error in the computed azimuth under one minute.

When the time of observation is not accurately known the altitude method should be used. If accurate time is known the hour angle method of determining the sun’s azimuth will give the best results.

Equal Altitude Method

The determination of azimuth by equal altitudes of the sun is very similar to the Indian Circle method discussed in Chapter L-

Both methods are based on the theory that the sun’s center at equal altitudes occupies symmetrical positions in azimuth east and west of the meridian in the morning and in the afternoon. If the declination

of the sun is considered constant 0 its path is symmetrical with respect to the meridian. In this case the meridian would be midway between any two positions of the sun which are at equal distance above the horizon. Unlike the Indian Circle method@ azimuth determined by the

Equal Altitude method is corrected for the error introduced by the change in the sun’s declination in the interval between the A.M. and

P 0 M, observations.

The field procedure used in the Equal Altitude method is 91 quite simple. With the transit thoroughly leveled a horizontal angle is turned from a fixed reference point to the sun’s center. The horizontal and vertical angle and the time of observation are read and recorded. At intervals of four or five minutes the above operation is repeated at least two more times. This constitutes the morning observation.

In the afternoon the transit is again carefully leveled. With the scope in the same position the largest vertical angle observed in the morning is set off and the sun is tracked until this position is reached by the center of the sun. The horizontal angle and the time are then recorded and the next vertical angle is set off and the procedure repeated.

If the transit has been carefully leveled then any other error introduced by poor instrument adjustment need not be considered since the vertical angles are identical in both the A.M. and P.M. obser vations . The same is true of both refraction and parallax and no correction is necessary.

The correction (C), expressed in minutes of angular measure, due to the change in declination is

n - 1/2 A Dec, tt,i\ V 600606009060 0 OQOOOeOOOOOOeeeOOOOOOO cos Phi. x sin t

In this equation A Dec, is the change in declination of the sun from the A.M. to the P.M. observation expressed in minutes of angular measure. The solution of Eq. (47) is simplified by substituting 92

l/2(Ti f Tg) for the hour angle (t), This gives

1 / 2 A Dec o L " cos~ Phi<= x sin 1/2(T^ T2) •••••••••••••••••••••••■• where (Tjl + Tg) is the total watch time from the A.M. to the P.M., observationg expressed in angular measure.

The correction (C) is applied after the horizontal angle between the sun8s position in the morning and afternoon is bisected.

The true south point is obtained by applying the correction to the east with a northerly or positive hourly change in declination, or to the west with a southerly or negative hourly change,

Table 22 of the Standard Field Tables aids in the solution of

Eq, ( 4 8 ) Using arguments of Phi. and 1/2(T^ t Tg) a coefficient can be taken directly from the table. Applying this coefficient to

1/2 ADec. gives the required correction (C)=

This method is seldom used by the average surveyor due. to the inconvenience of making observations both in the morning and afternoon and the latter at a precise time. It is a convenient method of establishing a true meridian at a base camp since it does not require an accurate measurement of either time or latitude.

The most accurate results are obtained when the sun is moving rapidly in altitude.

^•United States Department of the Interior, Bureau of Land Management, Standard Field Tables, 1956, page 210, CHAPTER IX

COMPUTATIONS

Introduction^

The equations discussed in the preceding pages can be solved in a number of ways* The most satisfactory method is one that will give the desired accuracy with a minimum of time and effort* There should be a continual regard for economy by keeping the computing device consistent with the precision of the equipment used* A

10-inch slide rule should not be used to compute the azimuth of a line when a directional theodolite was used to measure the angles 6

Computations should be made in an orderly and systematic fashiono The use of a standard fpnn adds to the efficiency and makes it easier for someone else to check the computationse

Slide Rule

The standard 10-inch rule can be read to three significant figures and is not precise enough for most azimuth computations/ A

20-inch slide rule that was especially designed for the Bureau of

Land Management gives much better results* One side of the rule is used for stadia reduction while the other side is laid out for solving the azimuth equation* 94

Like all logarithmic scales the accuracy of reading is dependent on the location of the number being read on the scale« Most of the functions can be read within two minutes of arc. This slide rule is not intended for use where an accurate azimuth determination is needed. It does offer a rapid method of finding the approximate bearing of a line.

Logarithms

The use of logarithms provide a convenient method of solving the azimuth equations. Pocket sized tables such as those published by

Lefax afford an excellent method of computing azimuth in the field.

Since addition and subtraction are the only operations requiredg the computations can be made in a very orderly fashion. Plate 3 is an excellent example of the simplicity of this method.

Five place logs will satisfy most azimuth determinations by the altitude method. When the value of the vertical angle is known only to the nearest minute there is little justification in using more than five places. Seven place tables are readily available for more accurate work.where the azimuth is to be determined within a few seconds of arc.

•hfon Vega„ op.cit. Natural Functions

The natural trigometrie functions are probably the most commonly used method of solving the azimuth equations. Used in conjunction with a desk calculator they offer a fast and accurate means of solving any of the solar computations. Division and multiplication can be done very rapidly on a modern electric calcu lator and answers obtained to ten or twelve significant figures,

A number of tables are available that can be used to find the natural trigometrie functions. One of the most convenient to use is Peter's eight place tables for each second of.arc,^

A small compact calculator for field work has recently been developed by the Curta Company, These pocket-sized calculators can be used to multiply and divide numbers with up to ten significant figures,.

/ Electronic Digital Computer

The use of a modern electronic digital computer is the fastest method of handling a large number of azimuth computations. Once the program is set up any of the equations used in determining azimuth can be solved in seconds.

Many surveyors have only an occasional need for solar obser-

•^Professor Dr, J, Peters$ Eight Place Table of Trigonometric

Functions 9 Edward Brothers , Inc, 8 Ann Az^or% 'Michigan, 1943, vat ions and the time required for the computations is -relatively

unimportant. Larger engineering firms, and some of our government

agencies, that are involved in mapping, take many solar observations

and the problem of computing azimuth becomes a significant factor.

Field work in our northern states and much of the mountainous area

throughout the country is restricted to the summer months. The end

of the field season finds some of our government agencies with a

backlog of solar observations numbering in the hundreds. The use of

an electronic digital computer may well be justified.

The principle disadvantages of using a computer are the

rather lengthy initial time required to write a satisfactory program

and the high cost of machine rental time. There are several advantages

that may well offset the disadvantages. Once the. program has been

written there is no need for an engineer to make any of the computa

tions. The required data can be entered directly from the field book

by anyone trained to punch I.B.M, cards. Once the correct data has

" been punched on the cards the possibility of error is very slight.

As was mentioned before this method is the fastest available to handle

a large number of computations. One other important advantage is that

each solar observation can be treated independently. This gives much

better results than using the average value of several observations. Hour Angle Program

A program was written to solve the hour angle equation (35) by means of an electronic digital computer<, The hour angle method was selected to program because it is the most accurate method of deter mining azimuth by the sun. The program was written using the Fortran language. Explanations of the program will be limited to the procedure for entering the required data and reading the print out sheet. No attempt will be made to explain the Fortran language as this can be obtained from any Fortran manual.^

The program exactly as it appears on Fortran 80 column statement cards is as follows.

^■Robert E, Smith and Dora E. Johnson $ Fortran Autotester

John Wiley and Sons 8 Inc, 9 New York e N.Y, g 19627"~='“’="=’ ***. MURPHY

* COMPILE FORTRAN 9 EXECUTE FORTRAN

C AZIMUTH BY HOUR ANGLE OF THE SUN C JERRY MURHPY DIMENSION D(4 ) ,DM(4) ,DS(4),H(3 ) ,HM(3)SRAD(4 ) ,RADT(3),HS(3) PI = 3.1415926 1 READ 31, MON, ID,IYR 31 FORMAT (312) GO TO (50,51,52,53,54,55,56,57,58,59,60,61),MON 50 PRINT 6 2 9ID,IYR

GO TO 2 0 0 51 PRINT 63, ID, IYR

GO TO 2 0 0 52 PRINT 64, ID, IYR

GO TO 2 0 0 53 PRINT 65, ID, IYR

GO TO 2 0 0

54 PRINT 6 6 ,ID,IYR

GO TO 2 0 0 55 PRINT 67, ID, IYR

GO TO 2 0 0

56 PRINT 6 8 , ID, IYR

GO TO 2 0 0 57 PRINT 69, ID, IYR

GO TO 2 0 0 58 PRINT 70, ID, IYR

GO TO 2 0 0 59 PRINT 71, ID, IYR

GO TO 2 0 0 60 PRINT 72, ID,IYR

GO TO 2 0 0 61 PRINT 73, ID, IYR 62 FORMAT (///9H JANUARY , 12, 4H, 19,12) 63 FORMAT (/// 10H FEBRUARY , 12, 4H, 19,12) non ooooooo 0 FRA (10F8.0) FORMAT 205 0 FRA (F8.0) FORMAT 207 HS(3) 207, READ 400 ,DM(3),DS(3),D(4) ) ,D(2),DM(2),DS(2),D(3 ) ,DS(1 ) ,DM(1 ) 205,D(1 READ 200 0 FRA (10F8.0) FORMAT 206 0 RA 0,M4),S4),()H()H()H2),M2),S2,( ,HM(3) ) ,HS(2),H(3 ) ,HM(2 ) ,H(1),HM(1),HS(1),H(2 ) ,DS(4 ) 206,DM(4 READ 300 4FRA (/7 MRH 1,4HS 19, 12) S H ,12, 4 MARCH (///7H FORMAT 64 7 OMT (/// FORMAT 67 19, 12) ,12, 4H, APRIL (///7H FORMAT 65 9 OMT (/// FORMAT 69 8 6 6 6 2 OMT //0 NVME ,1, H 19, 12) 4H, , 12, NOVEMBER 12) 19, (///10H 4H, , 12, FORMAT 72 12) , 19, OCTOBER H 4 (///9H , 12, FORMAT 71 SEPTEMBER (///11H FORMAT 70 3 OMT //0 DCME ,1, H 19, 12) 4H, , 12, DECEMBER (///10H FORMAT 73 6 RD( =,6799( H(I)+HM(I)/60.+HS(I)/3600.) )=0,26179939*( RADT(I 4 0.01745329*(D(I)+DM(I)/60.+DS(l‘)/3600.) = RAD(I) 3 5 READ READ 5 OMT //H A , 2 4H$ 9 12) $ 19, H 4 12, , MAY (///5H FORMAT OMT (/// FORMAT OMT 680 211) (6F8.0, FORMAT AI = 0.26179939*(THR+TMIN/60.+TSEC/3600,) = RADIT FNX CR DE NTCNANEHMRS NOPAE N O 50 COL IN 1 PLACE INFO 50 EPHEMERIS COL IN CONTAIN 2 NOT PLACE DOES INFO CARD NEXT EPHEMERIS IF 49 CONTAINS COL CARD IN 1 NEXT IF PLACE INVERTED IS INSTR IF RADIANS IN TIME OF EQU = RADIANS IN RADT(l) DECL, IN CHANGE = RAD(4) RADIANS IN LONGITUDE = RADIANS RAD(2) IN LATITUDE = RAD(l) 1=1,3 4 DO 1=1,4 DOS AI 0.01745329*(DEG+PMIN/60.+SEC/3600.) = RADI RADIANS IN ZONE RADIANS IN TIME = TIME OF RADT(3) EQU IN CHANGE = RADT(2) RADIANS IN DECLINATION = RAD(3) 6 , DEG, PMIN, SEC, THR, THIN, TSEC, INV, NEXT INV, TSEC, THIN, THR, SEC, PMIN, , DEG, 6 6 8 JN , 2 4, 9 12) 19, 4H, ,12, JUNE H JL , 2 4, 9 12) 19, 4H, ,12, JULY H AGS , 2 4H , 12* s 19., 12) AUGUST H

CD CD GMT = RADII + RADIO)' I = GMT~RAD(2) + RADT(1)+RADT(2)*GMT*3.8197186-PI DEC = RAD(3) + RAD(A)*GMT*3,8197186 COTZ = (SINF(RAD( 1 ))*COSF(T)-COSF(RAD(1)* SINF(DEC)/COSF(DEC)) 1/SINF(T) IF(ABSF(C0TZ)-1oE-7)7$7 98

8 TA - 1./C0TZ Z = ATAMF(TA) IF(RADIT-PI)9$10S10 9 IF(Z)11,12,12 11 B = Z+RADI+PI GO TO 13 12 B = Z+RADI GO TO 13 10 IF(Z)1U,11,11 14 B = 2.*PI+Z+RADI 13 BDEG = 6*57,295779 IF (BDEG-360,) 301,301,302 302 BDEG = BDEG - 360, 301 IDEG = BDEG A = IDEG C = (BDEG-A)*60, IMIN = C A = IMIN SEC = (C-A)*60, IF(INV)15,16,15 16 PRINT 17, IDEG, IMIN, SEC 17 FORMAT (14,2H -,13,2H -,F5,1) GO TO (5,1),NEXT 15 PRINT 18, IDEG, IMIN, SEC 18 FORMAT (14,2H -,13,2H -,F5.1, 9H INVERTED) GO TO (5,1), NEXT 7 Z = PI/2, IF(RADIT-PI) 19,20 ,20

19 IF(Z)21,22,22 100 21 B = Z+RADH-P1 GO TO 23 22 B = Z+RADI GO TO 23

2 0 IF(Z)24$21s21 24 B = 2.*PJ+Z+RADI 23 BDEG = B*57.295779 IF (BDEG-360») 201,201*202 202 BDEG = BDEG-360« 201 IDEG = BDEG IDEG = BDEG A = IDEG C = (BDEG-A)*60. IMIN - C A = IMIN SEC = (C-A)*60. IF(INV)25,26,25 26 PRINT 27, IDEG, IMIN, SEC 27 FORMAT(14,2H -,13,2H -,F5.1, 19H Z ASSUMED = 90 DEG ) GO TO (5,1), NEXT 25 PRINT 28, IDEG, IMIN, SEC 28 FORMAT(14,2H -,13,2H -,F5«1S 9H INVERTED, 19H Z ASSUMED = 90 DEG) GO TO (5,1).NEXT END 1 0 1 102 -

The statement cards are followed by a blank card and then the data cards,

The first data card refers to statement (1) and contains

the monthd day 9 and year of the observations. This information is entered in the first six columns of the statement card without the use

of decimal points. Two columns for the month 9 two for the day, and two for the year, June 13g 1963 would be entered as 061363,

The second data card contains the information asked for in statement (200), This data is entered in fields of eight on the statement card and requires the use of a decimal point. Use of a fixed number of columns for entering the data permits use of a • control card that greatly aids the key punch operator. The infor mation on this card is as follows:

Column No, Data

1 - 8 6 ,Latitude of station, degrees -

9 - 16 Latitude of statione minutes

17 - 24 ,Latitude.of station, seconds

25 - 32 « , Longitude of station * degrees

33 - 40 ...... Longitude of station, minutes

41 - 48 ,,,,,,,,,,,,,Longitude of station, seconds

49 - 56 ,Sun8s Declination, degrees

57 - 64 eSmVs Declination, minutes

65 - 72 Sun8s Declination, seconds

73 - 80 Change in Dec, per h r , , degrees The third data card refers to statement number (300)„

All data is entered in fields of eight and requires the use of a decimal pointc

Column Noo Data

1 - 8 ««oa,,*ee,ao,oe.*Change in Dec* per hr0p minutes

9 ~ 16 Change in Dec® per hr 0 g seconds

17 « 24 ,@«e**o*.«,oa,soEquation of Time % hours

25 - 32 o b ««e•«e a«•.••eEquation of T ime 9 minutes

33 - 40 a a «» , « * 0 a a a a a a a 0 eEqUBtiOD Of TilUQ % SeCOndS

41 - 48 e Change in E q e of Time per hro 9 hours

49 - 56 6 6 « 3 8a e e»e o6 e e o Change in Eq<, of Time per hr 0 9 minutes

57 « 64 o o o « 0 e«o 6 »e 6 € 6 eChange in E q & of Time per hr 0 9 seconds

65 - 72 * * e *«„,@ cTime Zone9 hours

73 - 80 *e*Time Zone 9 minutes

The fourth data card contains only the time zone in seconds

as called for in statement number (400) 6 This is placed in the first eight columns and requires a decimal pointe

The information on the first four cards will be the same for all observations taken on the same day and at the same station«

The date and time zone will be known 0 The latitude and longitude of the observing station would be scaled from a map * The remaining information on the first four cards is taken from the ephemeris for the date of the observation* 104

Data card number five contains the information from an individual observation. The horizontal angle and time of obser™ vation are entered in fields of eight and require the use of decimal points. The horizontal angle recorded must be the clockwise angle from the sun to the target. The time of observation is measured in the standard time for the time belt of the observing station, -

Column No. Data

1 « 8 ,.Horizontal Angle$ degrees .

9 ■= 16 * ...... Horizontal Angle 9 minutes

17 - 24 .Horizontal Anglee seconds

25 - 32 .Time of Observation, hours

33 - 40 ,Time of Observation» minutes

41 - 48 .Time of Observation, seconds

50 ..o,.,.,o,,,, 8 ...,...oControl Statement

Column (49) is left blank if the telescope was direct and contains a (1) if the instrument was inverted. Column number (50) is a program control. If column (50) contains a (1) then another observation will follow. If column (50) contains a.

(2 ) the program will return to read statement ( 1 ) and read a new set of ephemeris data. A decimal point is hot used in either column

(49) or (50), 105

A separate data card is used for each individual observation<,

The total number of data cards is four plus the number of observations.

If four observations were made at the same station then data cards

(5)e (6 )g (7)g and ( 8 ) would contain the horizontal angle and standard time of each observation. Column number (49) would indicate the position of the telescope, whether direct or inverted, for each

observation. On data cards number (5), (6 ), and (7) column number

(50) would contain a ( 1 ), On data card number ( 8 ) column number

(50) would contain a (2) if another series of observations were to follow taken either from a different station or on another date.

The print out sheet will contain the date of the observation.

Following the date will be the computed azimuth for each observation.

The computed azimuth will be the clockwise angle reckoned from the north. If the instrument was inverted the azimuth will be followed by the word "Inverted". CHAPTER X

CONCLUSION

The need for accurate determination of direction is not a thing of the past. The rapid growth of our country and increased value of land have added to the problems of the property surveyor.

Retracing old property lines and establishing new ones frequently require celestial observations. The survey of the public lands is far from complete. The addition of Alaska to the United States adds a considerable area to be subdivided by the Bureau of Land

Management.

The work of the Geological Survey continues over most of the country. Ground controls for both mapping and aerial photo graphs require high quality meridian determinations,

The guidance systems used in missiles and the controlling devices in our tracking stations are dependent on good direction

control. 11 The inertial guidance system used in missiles and submarines is capable of maintaining a direction within a few

seconds, but it must be reckoned 9 calibrated, and checked against reliable meridians".^

^Eldridge, op.cit., page 123.

106 107

The interstate highway system and the vast network of pipe lines and powerlines that cross this country depend on quality direction control*

Solar observations are frequently the most practical and

economical method of establishing the "true" meridian 6 The principal advantage of using the sun in determining azimuth is that observation can be made in the daytime and usually during regular working hours*

The size and brightness of the sun make pointing difficult and are the major drawbacks in solar observations *

There are several methods of pointing a telescope at the sun*

The least accurate methods employ a screen and sightings are made on a reflected image* Best results are obtained by a direct sighting on the sun* Use of a solar filter* either alone or in combination with a solar reticle * provides an accurate means of pointing directly on the sun* The most recent and;.re fined method of

pointing is by use of the Roelofs 8 solar prism*

High accuracy meridian, determinations require that each observation be used independently to compute azimuth* Use of the average values of several observations made on the sun?s limbs are subject to two systematic errors due to curvature and seraidiamefer«

The sun?s declination is given in the ephemeris and the tabulated value can be corrected to any desired accuracy*

Latitude and longitude are normally scaled from a map for solar observation made in the United States* Values within a few 10 8

seconds of arc can be obtained quite easily 6

Several methods can be used to determine azimuth by the sun*

Only two of these* the altitude and hour angle methods* are of practical value to the average surveyor*

The altitude method* as the name implies* requires an

accurate measurement of the sun*s altitude 0 A number of factors af fect the measurement of this quantity* The precision of the transit and atmospheric refraction are probably the most significant* The dB/dh ratio is the most important criterion as to the accuracy of the altitude method*

The hour angle method requires an accurate measurement of the local hour angle of the sun* Three factors affect the hour angle; the time of observation * longitude of the observing station* and the equation of time* In well mapped areas it is only the time of observation that is difficult to determine* The dB/dt ratio is the most important consideration when determining azimuth by the hour angle method*

The first and most significant consideration in choosing between the altitude and hour angle method is the precision of the survey timepiece* When accurate time control is available the hour angle method should be used*

Quality azimuth determinations can be made by use of the sun* Consistent and accurate results-demand a thorough knowledge of the principles involved* BIBLIOGRAPHY

Books

Breed^vQharles B.g Surveyinga John Wiley and Sons„ Inc. „

New York 9 1942.

Breedj Charles B.$ and Hosmer„ George L.e The Principles And. Practice of Surveying* Volume I - Elementary Surveying

Brown, Curtis M,„ and Eldridge 6 Winfield H.e Evidence And Procedures

for Boundary Location 9 John Wiley and Sons, Inc.a New,York 9 . : 1962.

Davis* Raymond E. „ and Foote * Francis S., Surveyings. Theory And

y Practice 9 McGraw-Hill Book Company* Inc.„ New York * 4th ed0* : 1953.

, . Gurley* W. and L. E.* The Gurley Telescopic Solar Transits Its Use and Adjuatment"*' ¥uiiWtln ^o'r, ii^»T*' Troy a New York * 195^.

Hart y William L.* College Trigonometry * D. C. Heath and Company* Boston* 1951.

Hiekerson* T. F.* Latitude* Longitude* and Azimuth by The Sun or . Stars * published by the Author * Chapel Hill *' N . dV * 19477

Kiely* Edmond R,* Surveying Instruments; Their History and ■ - Classroom Use * Bureau ofL Piibiications * Teachers College * ColuSbia tiniversity» New York* 1947.

Linsley* Ray K.* Kohler* Max A.* and Paulhus* Joseph L. H.* ■ Hydrology For Engineers * McGraw-Hill Book Company * Inc.*

Ngw York*' '1'96 '8 .""" ' ' .

Nasati* Jason John * A Textbook of Practical Astronomy* McGraw-Hill Book Company * New York * 1932.

Peters * Dr. J. * Eight Place Tables of Trigonometric Functions *'Edward Brothers%Inc% * Ann Arbor* Michigan* 1943.

109 110

Roelofs9 R,9 Astronomy Applied to Land Surveying* N. Vc WecL J» Ahrend and Zoc^t AwteMaAMollahd','' "&50."

Smith 9 Robert E«, „ and Johnson 9 Dora E., Fortran Autotester6 John Wiley and Sons, Inc.@ Hew York $ 1962. ~.... .

U. S. Bureau of Land Management9 Manual of Instructions For The Survey of The Public Lands of The United States 1947„ GoWram^T^i^ting O ^ i c e j, Washington 25 9 D. C.

U. S. Bureau of Land Managements Standard Field Tables, Government Printing Office „ Washingtw™2ET^°7rT"

U. S. Bureau of Land Management, Sun. Polaris, and Other Selected Stars a Government Printing Office 9 Washihgt'on^l^r^^ C.

U. S. Coast and Geodetic Survey» Special Publication Ho. 14» Government Printing OfficeV Washihgton 25, B. C. .

U. S. Coast and Geodetic Survey, Special Publication Ho. 247, Government Printing Office^^asHTngtm^T^BT’^cT^-”"

U. S. Geologic Survey, Topographic Instruction, Solar Observations .por Transit T r a v e r s ^ Government Printing' Off^ice, Washington 1953. •

-U. S. Wavy$ Tables of Computed Altitude and Azimuth, Hydrographic Office Publication No. 2'l4'^ ^overn'm%t'' Printing Office „ Washington 25, D, C.

Vega, Baron Von, Seven Place Logarithmic Tables. D, Van Hostrand Company, Inc., Princetons Hew Jersey.

Journals 9 Proceedings 8 and Transactions

Berry s Ralph Moor®„ "Azimuth by Solar-Altitude Observations How Good Is It?", Surveying And Mapping, Volume XVIIIs No. 3S July-Sept«~ 1958s American Congress On Surveying And Mapping, Washington6 D. C.

Eldridge9 Winfield H., "Purpose and Procedures for Meridian ' Determinations"s Proceedings, Illinois Land Surveyors Oonf. * RLSA* V o l . T i l , 'Urban, 111. 9 1962. Eldridges Winfield Ho a "Discussion of Solar-Altitude Azimuth"e Proceedings of Surveying and Mapping Division, No. 3410,

Hartman8 Paul, "Solar-Altitude Azimuth", Journal of the Surveying and Mapping Division, No. SU1, AScis, Feb. , 1963.

Bickerson, T. F.,"Determination of Position and Azimuth by Simple and Accurate Methods", Transactions of ASCE, Vol. 114, 1949,

Inch, Philip L., "Simplified Method of Determining True Bearing", Transaction of ASCE, Vol. 102, 1937.

Kriegh, James D., "Solar Observations For Determining The Bearing of A Line, How Good Are They?", Proceedings of The Sixth Arizona Land Surveyors* Converence, Ehglheerlng Experiment