Adjustment Computations Adjustment Computations: Spatial Data Analysis

Chapter 7

Astronomical Observations for Azimuth

Introduction

For years surveyors have sighted celestial objects to determine geodetic positions of stations and directions of lines. Due to the increased availability of control monuments and the Global Positioning System (GPS), the need for astronomical work has greatly diminished. Yet even today astronomical observations for azimuth can help the boundary surveyor determine a reproducable direction of a boundary for future resurveys. This worksheet looks at the celestial coordinate system and the development of equations for the determination of azimuths.

Development of Hour-angle Equation

Figure 1 shows the Celestial Coordinate System. The system is a heliocentric system. However since the distance to the nearest star is 109 times farther than the sun-earth distance, the sun-earth system is considered to be a point at the center of the celestial sphere with all stars motionless on the sphere. Thus the star's create a stable reference system for observations from the earth.

The Celestial Ephermis Pole (CEP) is the intersection of the earth's instantaneous spin axis with the celestial sphere. The celestial equator is the intersection of the earth's instantaneous equatorial plane with the celestial sphere. The vernal point ^ is the location on the celestial equator where the sun goes from the southern hemisphere to the northern hemisphere; the first day of spring. This location defines the x axis of the right ascension coordinate system. In this system the position of the star is given by its declination d and right ascension ^. However surveyors often use the Greenwich Hour Angle (GHA). This is the angle in the instantaneous equatorial plane from the Greenwich meridian (GM) to the meridian containing the star. This defines the "longitude" of the star at the time of observation.

Figure 1 Components of the celestial coordinate system.

Zenith is the point on the celestial sphere that is directly above the observer and opposite the direction of gravity. It is located by the observer's astronomical latitude F and longitude L. The Local Hour Angle (LHA) is the angle in the celestial equator from the observer's meridian to the meridian of the star.

The locations of the CEP (P), zenith of the observer Z, and the star (S) form the PZS spherical triangle. The arc length from the zenith to the pole is known as the co-latitude and is 90° - F. The arc length from the star to the pole is known as the co-declination and is 90° - d. The arc length from the observer's zenith to the star is known as the co-altitude and is 90° - h where h is the altitude angle to the star. The interior angle at the pole is known as the hour angle, t.

As can be seen in Figure 1, the line from zenith Z to the pole P defines north. Since the Earth rotates about its axis each day, the size of angle t increases until if reaches 180°. At this point PZS triangle flips to the other side of the of the observer's meridian. Thus the azimuth to the star is a function of the z angle. In developing and expression for determining the z angle, the altitude angle h should be removed from the expression since the line of sight from the observer to the star passes through the Earth's atmosphere. Thus refraction is a major systematic error in the observation of the altitude angle. With this in mind we will use spherical trigonometry to develop an expression for z that is not dependent on h.

Using the cosine law for sides yields

From the sine law, we have

Dividing Equation (2) by (1), we have

From the cosine law for sides, we can write the expression

Substituting Equation (4) into Equation (3) yields

Dividing both the numerator and denominator of Equation (5) by cosF cosd yields

Equation (6) is the so-called hour-angle formula. Notice that it is based soley on the hour angle t of the star, its declination d, and the astronomical latitude F of the observer. Since this function is not sensitive to small errors in latitude, it is sufficient to scale the latitude of the observing station from a map for most surveying purposes. As was noted previously, the t angle will go from zero to 180° before the triangle rotates to the other side of the observer's meridian. For western longitudes, Equation (6) can be written in terms of the LHA as

where the LHA is computed as

LHA = GHA + L

The relationship between the azimuth to the star, the LHA, and the z angle is shown in the following table.

When the LHA is if z is greater 0 if z is less than 0

0° to 180° Az = z + 180° Az = 360° + z

180° to 360° Az = z Az = 180° + z

Equation (7) is a function of time since both the GHA and declination of the star d are tabulated with respect to universal time (UT1). The value of UT1 can be determined from universal coordinated time (UTC) using the DUT correction. Both universal coordinated time and the DUT correction are broadcast in the United States by the National Institute of Standards and Technology radio station WWV at frequencies of 2.5, 5, 10, 15, and 20 MHz. These broadcast can also be heard by calling 1-3003-499-7111. In Canada, eastern standard time (EST) is broadcast by the radio station CHU at 3.33, 7.335, and 14.67 MHZ.

UT1 is based on the actual rotation of the earth. It is obtained by adding the DUT correction to UTC. The DUT correction is given by faint double ticks following the minute tone. For each double tick heard during the first seven seconds after the minute, an additional 0.1 seconds is added to UTC time heard at the minute tone to obtain UT1. If double ticks are heard between nine seconds and fifteen seconds, then an additonal 0.1 seconds must be subtracted from the UTC time heard at the minute tone. Thus UT1 is computed as


UT1 = UTC + DUT


Using the UT1 time and the appropriate date, the GHA and declination for the beginning of the day of the observation and the following day are obtained from ephemeris tables. The actual GHA and declination at the time of the observation are interpolated from the tables using the equations.

Once the azimuth to the star is determined, this azimuth must be transferred from the star to a line on the ground. The relationship between the ground point, horizontal angle and azimuth to the star is shown in Figure 2. To determine the azimuth of the line on the ground we add the explement of the horizontal angle to the azimuth of the star, or

A = Az + 360° - Hor. Angle

The following is the reduction of the observations in Example 18-2 in Elementary Surveying: An Introduction to Geomatics.

Figure 2 Relationship between

horizontal angle and azimuth to star.

Creating Functions

In this section the previously derived equations are converted into functional relationships.

Example

What is the azimuth of the line given the following observations on Polaris?

Day of Observation: 3 Dec, 2000

SOLUTION

//Correct the horizontal angles measured in Face II)

Date of observation in universal time.

Greenwich date of observation: 4 Dec, 2000

Ephemeris data:

Interpolate ephemeris data at time of observation

Compute the z angle

 

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