Astronomical Observations for Azimuth By: Charles D ...

Astronomical Observations for Azimuth By: Charles D. Ghilani, Ph.D.

Professor Emeritus of Engineering The Pennsylvania State University

1'st Edition

Copyright ? 1995, 2020 by Charles D. Ghilani All rights reserved Reproduction or translation of any part of this work beyond that permitted by the United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the author.

TABLE OF CONTENTS

USES OF CELESTIAL OBSERVATIONS FOR AZIMUTH . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

JUST WHICH NORTH ARE YOU TALKING ABOUT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

HISTORICAL METHODS OF DETERMINING AZIMUTH . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

ACCURATE METHODS OF AZIMUTH DETERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . 3

BASIC DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

SPHERICAL TRIGONOMETRIC FORMULAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

DERIVATION OF HOUR-ANGLE FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

SPECIAL EQUIPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

WHAT'S IN A CELESTIAL OBSERVATION? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 METHODS OF OBSERVING A CELESTIAL OBJECT FOR AZIMUTH . . . . . . . . . . 9 Universal Coordinated Time (9) Observing a Star (10) Observing the Sun (11) Field Procedures (11) REDUCING CELESTIAL OBSERVATIONS FOR AZIMUTH . . . . . . . . . . . . . . . . . . 13 Declination (13) Greenwich Hour Angle (GHA) and the Local Hour Angle (LHA) (14) Reduction Sheets (15) SAMPLE COMPUTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Computations Using Software (19)

ERRORS IN CELESTIAL OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1. WHY MAKE ASTRONOMICAL OBSERVATIONS FOR AZIMUTH?

In a retracement survey, the land surveyor adheres to the fundamental principle of "following the footsteps of the original surveyor". Oftentimes, however, when creating a new subdivision of land, the surveyor fails to provide the measurement which are necessary to perpetuate their own work. While most surveyors will monument corners with artificial monuments, few will establish any kind of recoverable spatial orientation for the lines. It is not uncommon for surveyors to use adjoining property lines for the bearing basis on the plot. Thus while the accuracy of distance and angle measurements has increased, the directions of lines may still be based on compass readings from the 19'th century. How many deeds exist today where the bearings of the lines disagree? How many deeds have lines based on a composite of several adjoiners? Furthermore, how many deeds have their directional orientation based on a single record line?

In fact, the record evidence for these lines is continually being lost due to the natural disappearance of monuments caused by erosion, corrosion, and man-made events. Thus when the monuments of the lines become lost, they themselves become unrecoverable. In fact, a surveyor who finds only a single monument in a property survey is confronted with the problem of trying to establish the spatial orientation (bearing basis) for the property. Astronomical observations for azimuth not only provide a known basis for a line's orientation, but also a repeatable reference for future surveyors.

On large traverse surveys, astronomical observations for azimuth can also provide checks on angles. Once experienced with the techniques of making astronomical observations, a surveyor will able be to determine a line's astronomical azimuth within 10 minutes to an accuracy less than ?15". In large traverses these periodic angular checks will pay for themselves by reducing the time it takes to isolate and eliminate any angular measurement errors.

2. JUST WHICH NORTH ARE YOU TALKING ABOUT?

Directions of lines are traditionally based upon the size of an angular arc from a reference meridian called North. The direction of the reference meridian may be determined from existing monuments, magnetic directions, map projection coordinates, celestial observations, or the polar axis of the Earth. Each of these reference meridians are briefly discussed below.

Assumed North is based on the existence of two monumented locations. The direction of the line connecting these two monuments is arbitrarily defined as North, and assigned an azimuth of 0? 00' 00". While this method is expedient to use, it is lost as soon as either of the monuments lost. Thus, this method is generally limited to small independent surveys.

Magnetic North is defined by the pull of the earth's magnetic forces. Since the magnetic poles of the earth are constantly changing, the magnetic directions are also constantly changing. Furthermore local attractions to the compass needle are created by iron deposits and artificially created magnetic fields caused mostly by electric power lines. These various sources can cause the needle of compass to vary by as much as 8' per day. Thus while magnetic directions are easily measured, they do not have any permanence or repeatability.

Geodetic North is defined by the mean rotational axis of the earth. This directional basis is also known as geographic north. While this system is comparatively permanent in nature, it cannot be directly measured. Thus, it can only be used in conjunction with reference monuments that have the direction of the connecting line previously determined.

Grid North is a based upon a map projection system. It is mathematically related to geodetic north, and thus requires the same monumentation as geodetic north.

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ASTRONOMICAL OBSERVATIONS FOR AZIMUTH

Astronomical (Celestial) North is north based upon a projection of the earth's polar axis onto a celestial sphere. This reference meridian can be directly measured in the field. However due to geoidal fluctuations, corrections must be made for the local variations in the direction of gravity. This correction to celestial north is called the Laplace correction, and can vary in size from -10" to +10" in Pennsylvania. The National Geodetic Survey has created a program called GEOID that models this correction based on the latitude and longitude of the observing station.

This text reviews the methodology of making, reducing, and analyzing celestial observations to determine the astronomical azimuth of a line and its standard deviation. It is intended to be educational for a student in a survey program as well as the practicing professional. Included with this booklet is a DOS program that will reduce a set of celestial observations for azimuth. The author hopes that the readers of this book find the material contained herein useful.

3. HISTORICAL METHODS OF DETERMINING AZIMUTH

The determination of the azimuth of a line using astronomical observations was nothing new to the

ancients. In fact, two relatively simple procedures can be used to get the approximate azimuth of a

line which do not require the knowledge of any mathematics. These methods are known as the

shadow method and the equal-altitude method.

In the shadow method shown in Figure 1, a rod is

placed vertically in a level area of the ground. During the

period of a day, the end of the rod's shadow is marked at

regularly timed intervals. After marking the shadow's progress,

a rope is stretched from the center of the pole to the arc of the

shadow, and used to scribe an arc that intersects the shadow at

two places. By connecting the two points of intersection, chord

is defined for the circular arc defined by the rope. Finally, the Figure 1 Shadow method.

line from the center of the pole to the bisector of this chord lies

on the astronomic meridian, and thus defines astronomic north. The accuracy of this method in

defining astronomic north is approximately ?30' of arc.

In the equal-altitude method which is shown in Figure 2, the

altitude (vertical) angle to the sun is measured in the mid-morning. The

observer must then wait until mid-afternoon when the sun reaches the

same altitude. The bisector of the horizontal angle defined by these two

points of equal-altitude is the astronomic meridian at location of the

instrument. This meridian can also be defined by bisecting the chord that

is defined by an arc connecting these two points of equal-altitude. This

method is also accurate to within 30' of arc.

Figure 2 Equal

altitude method.

ASTRONOMICAL OBSERVATIONS FOR AZIMUTH

3

4. ACCURATE METHODS OF AZIMUTH DETERMINATION

4.1) Basic Concepts: In Figure 3, it can be seen that the azimuth of the star equals the azimuth of the line plus the horizontal angle. Thus the azimuth of the line equals the azimuth of the star minus the measured horizontal angle, or in equation form is:

(1)

where Azline is the azimuth of the line at the time the azimuth of the star is determined, Azi is the azimuth of the star, and ? to the right is the clockwise horizontal angle from the line to the star.

If the rotation of the earth is ignored, it is possible to imagine all stars (excluding the sun) to be motionless points of light in the sky. Furthermore, if all stars are assumed to be an infinite distance from the earth, it is possible to imagine that all stars lie on an invisible sphere. This imaginary sphere is known as the celestial sphere. From this sphere, equations that model the apparent positions of the stars in relation to the earth are derived.

Now due to the rotation of the earth, the stars actually appear to move counter-clockwise around the earth's north pole. This apparent motion of the star causes the horizontal angle to the star to change with time. Therefore to accurately determine the azimuth of the star, and thus a line on the ground, the specific time and horizontal angle to the star must be recorded.

5. BASIC DEFINITIONS

Before proceeding any further with this development, it is first necessary to define specific terms used in astronomy.

Upper culmination is the highest point of a star's apparent rotation in the sky.

Figure 3 The apparent motion of a star as viewed from an observer's position on the Earth.

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