This document discusses theories of errors in astronomical observations used for surveying. It provides definitions and procedures for determining azimuth, latitude, and longitude through astronomical observations of stars and other celestial bodies. While astronomical methods were formerly used for establishing directions and coordinates, they have now been replaced by GNSS surveys which provide coordinates directly from satellite positioning data. The document outlines historical astronomical techniques and their replacements by modern survey methods.

1. THEORY OF ERRORS IN
OBSERVATIONS
Dr. Mahmood Arshad
Assistant Professor,
Dept. of Mining Engineering,
Faculty of Earth Sciences and
Engineering,
University of Engineering & Technology,
Lahore.
smarshad@uet.edu.pk
Min-E-240 Surveying
Lecture 4 – Week 2

2. Astronomical Observations
 Observing positions of the sun or certain stars.
 To determine the direction of the astronomic meridian
(astronomic north).

3. Astronomic Azimuth
 The resulting azimuth was needed to
 establish directions of new property lines so parcels could be
adequately described;
 to retrace old property boundaries whose descriptions include
bearings that were determined by astronomical methods;
 to specify directions of tangents on route surveys;
 to orient map sheets; and
 for many other purposes.
 These procedures have today been replaced with GNSS
surveys where the coordinates of two points are established
on the ground using either static and kinematic GNSS
methods as discussed in Chapters 14 and 15.

4. Lat. and Long.
 The latitudes and longitudes of points can also be
determined by making astronomical observations.
 Seldom done today for two reasons:
 the field procedures and computations involved, especially for
longitude, are quite difficult and time consuming especially if
accurate results are expected and
 the use of the global navigation satellite systems has now
made the determination of latitudes and longitudes a rather
routine operation.

5. Astronomic Meridian
 Astronomic meridian at any point it is a line tangent to, and
in the plane of, the great circle which passes through the
point and the Earth’s north and south geographic poles.

6.  https://www.youtube.com/watch?v=XkoNZ6IQZKc
 https://www.youtube.com/watch?v=Rm0E9RjsUWE

8. Procedures for Astronomical Azimuth
 A total station is set up and leveled at one end of the line
whose azimuth is to be determined, like point A of Figure
C.2;
 The station at the line’s other end, like B of Figure C.2, is
carefully sighted and the instrument’s horizontal circle
indexed to 0°00’00”
 The telescope is turned clockwise and the star S carefully
sighted;
 The horizontal, and sometimes vertical, circles of the
instrument are read at the instant of pointing on the star;
 The precise time of pointing is recorded; and
 The horizontal angle is recorded from the reference mark to
the star, like 𝜃 angle of Figure C.2 from B to S.

9. Office Work
 Obtaining the precise location of the star in the heavens at
the instant sighted from an ephemeris (almanac of celestial
body positions);
 Computing the star’s azimuth (angle Z in Figure C.2) based
on the observed and ephemeris data; and
 Calculating the line’s azimuth by applying the measured
horizontal angle to the computed azimuth of the star as
𝛼 = 360° + 𝑍 − 𝜃
 The sun and, in the northern hemisphere, Polaris (the north
star) are almost always selected.
 In the southern hemisphere, the star Sigmus Octantis and
the stars in the constellation Southern Cross are commonly
used for astronomical observations.

10. Accuracies in Astronomical Azimuths
 The precision of the instrument used,
 Ability and experience of the observer,
 Weather conditions,
 Quality of the clock or chronometer used to measure the
time of sighting,
 Celestial body sighted and its position when observed, and
 Accuracy of ephemeris and other data available.

12. Definitions
 The zenith is located where a plumb line projected upward meets the
celestial sphere.
Stated differently, it is the point on the celestial sphere vertically above
the observer.
 The nadir is the point on the celestial sphere vertically beneath the
observer and exactly opposite the zenith.
 The north celestial pole is point P where the Earth’s rotational axis,
extended from the north geographic pole, intersects the celestial
sphere.
 The south celestial pole is point where the Earth’s rotational axis,
extended from the south geographic pole, intersects the celestial
sphere.
 A great circle is any circle on the celestial sphere whose plane passes
through the center of the sphere.
 A vertical circle is any great circle of the celestial sphere passing
through the zenith and nadir and represents the line of intersection of a
vertical plane with the celestial sphere.

13. Definitions – Cont’d
 The celestial equator is the great circle on the celestial sphere whose
plane is perpendicular to the axis of rotation of the Earth.
 An hour circle is any great circle on the celestial sphere that passes
through the north and south celestial poles.
They correspond to meridians (longitudinal lines) and are used to
observe hour angles.
 The horizon is a great circle on the celestial sphere whose plane is
perpendicular to the direction of the plumb line.
 A celestial meridian, interchangeably called local meridian, is that
unique hour circle containing the observer’s zenith.
It is both an hour circle and a vertical circle.
 A diurnal circle is the complete path of travel of the sun or a star in its
apparent daily orbit about the Earth.
 Lower culmination—the body’s position when it is exactly on the lower
branch of the celestial meridian;
 Eastern elongation—where the body is farthest east of the celestial
meridian with its hour circle and vertical circle perpendicular;
 upper culmination; and western elongation

14. Definitions – Cont’d
 An hour angle exists between a meridian of reference and the hour
circle passing through a celestial body.
It is measured by the angle at the pole between the meridian and hour
circle, or by the arc of the equator intercepted by those circles.
 The Greenwich hour angle (GHA) of a heavenly body at any instant of
time is the angle, measured westward, from the upper branch of the
meridian of Greenwich to the meridian over which the body is located at
that moment.
Local hour angle (LHA) is similar to GHA, except it is observed from the
upper branch of the observer’s celestial meridian.
 A meridian angle is like a local hour angle, except it is measured either
eastward or westward from the observer’s meridian, and thus its value
is always between 0° and 180°.
 The declination of a heavenly body is the angular distance (measured
along the hour circle) between the body and the equator; it is plus when
the body is north of the equator and minus when south of it.

15. Definitions – Cont’d
 The polar distance or codeclination of a body is 90° minus declination.
 The position of a heavenly body with respect to the Earth at any
moment may be given by its Greenwich hour angle and declination.
 The altitude of a heavenly body is its angular distance measured along
a vertical circle above the horizon.
 The coaltitude or zenith distance is 90° minus the altitude.
 The astronomical or PZS triangle is the spherical triangle whose
vertices are the pole P, zenith Z, and astronomical body S.
 The azimuth of a heavenly body is angle observed in the horizon plane,
clockwise from north or south point, to vertical circle through body.
 The latitude of an observer is the angular distance, measured along the
meridian, from the equator to the observer’s position.
 An equinox is commonly regarded as the instant of time when the plane
of Earth's equator passes through the center of the Sun. This occurs
twice each year: around 20 March and 23 September.
 The right ascension of a heavenly body is the angular distance
measured eastward from the hour circle through the vernal equinox to

16. Time
 Sidereal time: A sidereal day is the interval of time between
two successive upper culminations of the vernal equinox
over the same meridian.
At any location for any instant, it is equal to the local hour
angle of the vernal equinox.
 Apparent solar time: An apparent solar day is the interval of
time between two successive lower culminations of the sun.
Since the Earth revolves about the sun once a year, there is
one less day of solar time in a year than sidereal time.
Thus, the length of a sidereal day is shorter than a solar
day by approximately 3 min 56 sec.

17. Time – Cont’d
 Mean solar, or civil, time: This time is related to a fictitious sun, called
the “mean” sun, which is assumed to move at a uniform rate.
It is the basis for watch time and the 24-h day. The equation of time is
the difference between “apparent” solar and “mean” solar time.
Local apparent time is obtained by adding the equation of time to local
civil time.
 Standard time: This is the mean time at meridians 15° or 1 h apart,
measured eastward and westward from Greenwich.
Eastern Standard Time (EST) at the 75th meridian differs from universal
time (UT), or Greenwich civil time (GCT), by 5 h (earlier, since the sun
has not yet traveled from the meridian of Greenwich to the United
States).
Daylight saving time (DST) in any zone is equal to the standard time in
the adjacent zone to the east; thus, central daylight time is equivalent to
Eastern Standard Time.

19. Online Course for Celestial Measurements
 https://my.vanderbilt.edu/astronav/overview/