# Surveying Astronomy

## Terms

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Concerning the PZS triangle, colatitude is to latitude as polar distance is to which of the following?

(A) Declination
(B) Altitude
(C) Hour angle
(D) Zenith distance
Solution: The latitude of an observer is the complement of the side of the PZS triangle called the colatitude. The complement of the polar distance is the declination of a star.

THE PZS TRIANGLE

18. Which of the following correctly identifies the points of intersection of the three great circles that form the spherical triangle called the PZS triangle?

(A) P: parallactic
Z: zone
S: south
(B
Solution: The PZS triangle is formed by the arcs of three great circles that intersect at the pole, the observer's zenith, and the star. The pole may be either the north or south, and the star may be any celestial body.

GEOGRAPHICAL COORDINATES AND THEIR ELEMENTS 4-9

14. Which of the following statements correctly describes properties of the geographical coordinate known as longitude?

(A) Each meridian of longitude lies in the plane of a great circ
Solution: The plane of each meridian of longitude passes through the center of the earth. Therefore, each meridian is an are of a great circle. The distance between meridians along a parallel of latitude grows quite short near the poles.

Since there are 24 hours in a mean solar day and 360* of longitude around the earth, 15* equals one mean solar hour.

360*
24 hr = 15*

12. Which of the following statements correctly describes properties of the geographical coordinate known as latitude?

(A) Latitude is measured in degrees, minutes, and seconds north and south of the equator.
(B) Every position on the ear
Solution: A parallel of latitude is a small circle on the surface of the earth. Every point along a parallel of latitude has the same latitude. So it cannot be said that any position, except perhaps the north and south poles, has a latitude unique to itself. Latitude is measured in degrees, minutes, and seconds north and south of the equator.

. What is the declination of a star if that star passes through your zenith where your latitude is 35*N?

(A) 35*N
(B) 55*N
(C) 90*N
(D) 35*S
Solution: The angle between the celestial equator and the zenith of the observer is the latitude of the observer. Since declination of stars is also measured from the celestial equator along the stars' hour circles, the star in question must have a declination equal to that of the observer's latitude. Any star that passes through an observer's zenith must have a declination equal to the observer's latitude. This principle is the foundation of one technique for determining the latitude of a position.

4-12 Surveying ASTRONOMY

20. Which of the following arcs of the great circles, both complements of tile sides of the PZS triangle angle, are used to find the azimuth angle of a star using the hour angle method?

(A) Latitude and decl

Solution: The three sides of the PZS triangle are segments of three great circles. These segments are known as the colatitude, the polar distance, and the zenith distance. The polar distance is also known as the codeclination and the zenith distance is also known as the coaltitude. The two sides of the PZS triangle whose complements are critical to solving the azimuth of a star by the hour angle method are the colatitude and the polar distance. The complements are the latitude of the observer and the declination of the star.

The formula for the hour angle is

Sin t
Tan z = cos 0 tan 8 cos t

The latitude (complement of the colatitude) and the declination (complement of the polar distance are both essential to the formula's solution.

. What is the latitude of the place where the maximum altitude of the sun above the horizon achieved at any time of the
year is 23"26 1/2'

(A) 90*00'00"S
(B) 23*26'30"N
(C) 90*00'00"N
(D) Both A and C<

Solution: Since the maximum declination of the sun is 23'26 1/2' north and south of the celestial equator, any latitude less than 90' must witness an altitude greater than 23'26 1/2' ' at some time. The only place where the maximum altitude of the sun is the same as its maximum declination is where the celestial equator corresponds with the horizon-at one of the earth's poles.

GEOGRAPHICAL COORDINATES AND T14FT-P ELEMENTS

11. Which of the following terms describes a great circle on which every point is equidistant from the north and the South Pole?

(A) The prime meridian
(B) The hour circle
(C)
Solution: The terrestrial equator lies on a plane that passes through the earth's center and is perpendicular to the polar axis of the earth. These two properties assure that every point on the terrestrial equator must be equidistant from the two poles of the earth.

The angle between an observer's zenith and the celestial equator will always be equal to which of the following?

(A) 90d
(B) The observer's longitude
(C) The right ascension
(D) The observer's latitude
Solution: This problem involves one of the principles of the imaginary system known as the celestial sphere. Terrestrial positions are transferred to the celestial sphere by lines prolonged from the center of the earth to the surface of the celestial sphere. Therefore, the angular distance from the terrestrial equator to an observer's position on the earth, which is the observer's latitude, remains unchanged when the points between which the latitude is reckoned are transferred to the surface of the celestial sphere.

19. The angle between the great circles that form two of the sides of the PZS triangle is the azimuth of the star being observed. Which are those two circles?

(A) The prime vertical and the hour circle through the star
(B) The observer's
Solution: The PZS triangle has six elements, one of which is called the zenith angle. The zenith angle is the angle between the vertical circle through the star and the observer's meridian. Both the vertical circle through the star and the observer's meridian are thought of as great circles on the celestial sphere.

. Which value corresponds to the sum of a star's hour angle and its right ascension?

(A) The azimuth of the star
(B) The longitude of the observer
(C) The zenith distance of the star
(D) The sidereal time at the observer's posit
Solution: The right ascension of a star is measured in the opposite direction from the apparent rotation of the celestial sphere. Hour angles are measured in the same direction as the apparent rotation of the celestial sphere. Both values are measured to the hour circle of the star, but from different origins. The right ascension is from the First Point of Aries and the local hour angle is from the observer's meridian. Their sum is equal to the local sidereal time at the observer's position, since zero hour sidereal would have occurred when the First Point of Aries was last at the upper transit on the observer's meridian.

1. The spherical triangle known as the PZS triangle has sides that are segments of three great circles. Which two of those great circles intersect at the star?

(A) The prime vertical and the celestial equator
(B) The hour circle and the h
Solution. The acronym PZS stands for three points on the celestial sphere: the pole, the zenith, and the star. Each pair of points is included in a great circle. The great circle through the pole and the star is known as the hour circle. The great circle through the zenith and the star is known as the vertical circle. The great circle through the pole and the zenith is known as the observers meridian.

One sidereal hour after a star has set exactly in the west, the First Point of Aries stands precisely at upper transit on an observer's meridian. At that instant, what is the right ascension of the star?

(A) 255-00'
(B) 180-00'
(C) 1
Solution: Sidereal time, also known as star time, is the system used to measure the movement of the stars. While this system of time is not synchronous with solar time, there are nevertheless 24 sidereal hours in a sidereal day. Zero hour in local sidereal time occurs when the First Point of Aries, a point on the celestial equator, is at upper transit on an observer's meridian. Twenty-four sidereal hours later, the First Point of Aries once again stands at the same meridian. Since it must travel 360' in those 24 sidereal hours, it must travel 15' per sidereal hour.

The star that sets precisely in the west must also be on the celestial equator. As with all stars, its right ascension is measured from the First Point of Aries in the direction opposite the apparent rotation of the celestial sphere. Had the star in question set at the same moment that the First Point of Aries reached upper transit, its right ascension would be 270' (remember, right ascension is measured opposite the direction of apparent rotation). However, the star set one sidereal hour, or 15^ before the First Point of Aries reached upper transit. Therefore, the star's right ascension is 270' less 15^, or 255'.

. Standing at 40*N latitude, you observe the sun rise precisely in the east. What is the declination of the sun and what is the approximate date?

(A) Declination 0*, June 22 or Dec. 22
(B) Declination 23*26 1/2' June 22 or Dec. 22
(C
Solution: In order to rise exactly in the east, the sun must lie directly on the celestial equator. This occurs twice a year, on Mar. 21 (the vernal equinox) and Sept. 23 (the autumnal equinox). These dates vary somewhat from year to year because the exact times of the equinoxes vary.

Which astronomical coordinates are constantly changing and can readily be measured by a theodolite?

(A) Right ascension and declination
(B) Hour angle and declination
(C) Hour angle and parallactic angle
(D) Altitude and azimuth
Solution: The altitude and azimuth of a celestial body can be measured with respect to the observer's horizon and astronomic north. An instrument such as a theodolite can be successfully oriented to these references with relative ease. The references of the other astronomic coordinates include the First Point of Aries and the celestial equator, among others. It would be considerably more difficult to orient a theodolite to such references.

21. Which element of the PZS triangle is the least used in surveying astronomy?

(A) The zenith angle
(B) The hour angle
(C) The parallactic angle
(D) The colatitude
Solution: The parallactic angle, formed by the intersection of the vertical circle and the hour circle is the least-used element of the PZS triangle. While it