A comparison of the methods used in determining azimuth by solar observations Item Type text; Thesis-Reproduction (electronic) Authors Murphy, Gerald Edward, 1931- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 29/09/2021 20:46:26 Link to Item http://hdl.handle.net/10150/319786 A COMPARISON OF THE METHODS USED IN DETERMINING AZIMUTH BY SOLAR OBSERVATIONS by . 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®<a©»eeoo©eeoo©eieo©ovo©<aoGUG©c<5o»<i» X 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©©»©<eo#ee©oet30 9 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©©»<iQco 13 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<?»D» 28 111 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 By 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