Download SU3150-Astronomy - Michigan Technological University

SU3150 – PRINCIPLES OF GEODESY
Course Notes
By
Indra Wijayratne
Associate Professor of Surveying
Michigan Technological University
Houghton, MI 49931
Copyright  January 2003
Indra Wijayratne
"Education is not necessarily knowing so much. It is knowing where to go to find what you
need; recognizing it when you find it; and how to use it when you have found
it"(Hammerstorm 1981)
WHAT IS GEODESY?
Dictionary Definition:
"A branch of applied mathematics which determines by observations and computations the
exact positions of points and the figures and areas of large portions of the earth's surface,
the shape and the size of the earth and the variation of terrestrial gravity and magnetism."
A Text Book Definition:
"The discipline that deals with the measurement and representation of the earth, including
its gravity field, in a three dimensional time varying space"
AGU Definition:
"The science that determines the size and shape of the earth, the precise positions and
elevations of points, and lengths and directions of lines on the earth's surface, and the
variations of terrestrial gravity"
Functions of Geodesy
Establishing reference datums and coordinate systems for the definition of
 horizontal positions of points
 distances and directions between points
 elevations of points
Mathematical projections necessary to depict the datum surface on a map
Determination of geophysical properties such as the gravity field on or near the surface of
the earth, geoid (mean sea level) and deflection of vertical (plumbline)
Study and monitor the geo-dynamics phenomena such as ocean and earth tides, crustal
(tectonic) movements, polar motion, and the variations in earth rotation and gravity field
Observation Techniques in Geodesy
 Astronomical
 Terrestrial
 Space
Astronomic Observations
 Latitude, Longitude
 Azimuth
 Very Long Baseline Interferometry (VLBI)
 Observations necessary to monitor polar motion, precession, nutation, etc.
Terrestrial Observations
 Arc measurements (historic)
 Triangulation, Trilateration, Traversing
 Leveling
 Zenith or vertical angles
 Gravity
Space Based Observations
 Lunar laser ranging
 Satellite laser ranging
 Satellite positioning
 Satellite altimetry
Applications of Geodesy
 Surveying and mapping
 Defense
 Geophysical explorations
 Space explorations
 Communication, Navigation, etc.
Geodesy is the common link connecting
 Surveying
 Photogrammetry
 Cartography
 Geodynamics
 Geophysics
 Physics
Branches of geodesy
 Geometric geodesy
 Gravimetric (physical) geodesy
 Satellite geodesy
Geometric geodesy deals with the shape and size of earth, distance and direction of lines
on earth, reference datums, and coordinate systems
Gravimetric or physical geodesy is the science that studies geophysical and geodynamic
properties of earth, and includes earth gravity field and attractions of sun, moon and
planets
Study of satellite orbits, motion, perturbations, and satellite based positioning falls under
satellite geodesy
Why Should A Surveyor Study Geodesy?
Geodetic control surveys
 Understanding geodetic datums and coordinate systems, e.g. NAD-83, WGS-84
 Difference between geodetic and astronomic coordinates
 Different between geodetic and astronomic azimuth
 Azimuth change due to convergence of meridians
 Lengths of lines on the datum surface
 Reduction of measured lines to datum surface
Geodetic leveling
 Different datums (geoids), e.g. NGVD29, NAVD88
 Orthometric height and dynamic height
 Effect of gravity on leveling
GPS Surveys
 Satellite datum(s)
 GPS derived coordinates and baselines
 GPS derived orthometric heights (elevations
State Plane Coordinates
 Why is it needed?
 Definition and implementation
 Relevant computations
A knowledge of geodetic principles are needed in any survey that
 Covers a very large area or a distance
 Has to meet very high accuracy standards
 Uses specialized techniques such as GPS
History of Geodesy
 Flat earth concept (Homer, 9th Century B.C.)
 Idea of spherical earth (Pythagoras, Aristotle, Newton)
 First attempt at measurements by Eratosthenes (220 B.C.) in Egypt
 Later attempts by
- Poseidonius (135-150 B.C.)
- I-Hsing (China, 8th century A.D.)
- Caliph Al-Mamun (Arab, 820 A.D.)
 Method of triangulation by Dutch mathematician Snellius (1615-1620)
 Controversy over oblate spheroid vs. prolate spheroid (pumpkin vs. egg shape)
Spherical Trigonometry
Spherical Trigonometry is the branch of mathematics that is used for solving spherical
triangles
Great many problems in astronomy and geodesy where distances (arcs lengths) or
directions of lines (arcs) need to be computed are solved by the use of spherical
trigonometry
Spherical trigonometry is the basis for all computational formulas used in practical
astronomy used in navigation and surveying
Following definitions and properties of circles are needed in order to understand spherical
coordinates and solution of spherical triangles
Great circle
A circle having the same center as that of the sphere
A great circle is formed on a sphere by the intersection of the sphere with a plane passing
through the center of the sphere
Small circle
A circle (on the sphere) whose center does not coincide with the center of the sphere
Small circles are not important in spherical trigonometry but important in astronomy and
geodesy
Poles (of a great circle)
The ends of the diameter, which is perpendicular to the plane of the great circle in question
Points to note
The shortest distance between any two points on the surface of a sphere is along the arc of
a great circle passing through these two points
Any number of great circles can be drawn through a given point
All great circles passing through two poles intersect the great circle, to which the poles
belong to, at right angles
Spherical Triangle
A spherical triangle is formed by the intersection of three great circle arcs
This means that the sides of a spherical triangle are NOT straight lines but arcs of circles
The length of a side of a spherical triangle is the length of the spherical arc
Recall that the length of a spherical arc is equal to the product of the radius of the sphere
and the angle, in radians, subtended at the center of the sphere by the spherical arc
In spherical trigonometry, the radius of the sphere is taken as unity, and therefore, the
length of the arc becomes equal to the angle it subtends at the center
The value of this angle can be expressed in any convenient units, e.g. degrees, minutes
and seconds
When the magnitude of a side is expressed as the angle subtended at the center, the
radius of the circle becomes irrelevant for most calculations
The angle at any vertex of a spherical triangle is the angle formed by the tangents to the
two great circle arcs at that vertex
The sum of three angles at the vertices of a spherical triangle, unlike those of a plane
triangle, do not add up to 180
This sum always exceeds 180,and this excess over 180 is called spherical excess
The spherical excess of a spherical triangle is equal to the ratio of the surface area of the
triangle to the square of the radius of the sphere, in radian measure
That is, the spherical excess is
 = area
R2
This shows that the spherical excess too does not depend on the size of the sphere on
which the spherical triangle is formed, but rather, on the size of the angles subtended by
the sides
Formulas for Solving
Spherical Triangles
Even though spherical trigonometry uses a different set of formulas for solution of triangles,
they have some properties common with plane triangles
Sine formula
Sin A =
sin a
sin B =
sin b
sin C
sin c
Compare with sine formula in plane trigonometry
Sin A = sin B = sin C
a
b
c
Cosine formula
Cos a = cos b.cos c + sin b.sin c.cos A
Cos b = cos c.cos a + sin c.sin a.cos B
Cos c = cos a.cos b + sin a.sin b.cos C
Another form
Cos A = -cos B.cos C + sin B.sin C.cos a
Cos B = -cos C.cos A + sin C.sin A.cos b
Cos C = -cos A.cos B + sin A.sin B.cos c
The cosine formula of the sides is best when all three sides or two sides and included angle
are known
The cosine formula of angles is best when all three angles or two angles and adjacent side
are known
Cotangent formula
Cot c.sin a = cos a.cos B + sin B.cot C
Cot c.sin b = cos b.cos A + sin A.cot C
Cot a.sin b = cos b.cos C + sin C.cot A
Cot a.sin c = cos c.cos B + sin B.cot A
Cot b.sin c = cos c.cos A + sin A.cot B
Cot b sin a = cos a.cos C + sin C.cot B
Napier’s rules
sine of middle part = product of cosines of
opposite parts
= product of tangents of
adjacent parts
Spherical Coordinates (Chap. 1)
Spherical or curvilinear coordinates are used for defining points on a spherical surface
These coordinates are used in astronomy to indicate positions of stars and other celestial
objects, and in geodesy, to indicate positions of points on earth
Often, computations such as the distances (arc lengths) between points or directions of
lines on a spherical surface require the use of spherical coordinates
Spherical coordinates can be used to define points even when the surface deviates slightly
from a perfect sphere
Above is the case in geodesy where the points are defined on an ellipsoid rather than on a
sphere
Spherical coordinates use arc lengths (angular measure) along great circles as opposed to
linear measures along straight axes in plane coordinates
A point is defined by two coordinates each of which is an arc along a specific great circle
and the coordinate value is the angle subtended at the center of the sphere by the
corresponding arc
One of the great circles is termed the primary great circle and the other the secondary great
circle
Secondary great circle, almost always, is the great circle passing through the point to be
defined
The primary great circle passes through the pole of the secondary great circle, and vice
versa
This means that the primary and secondary great circles intersect each other at right
angles
Geographic Coordinates
Latitude and longitude are spherical coordinates used for defining positions of points on
earth
They are, in general, called geographic coordinates when the points are defined, and
relevant computations are done, by assuming the earth to be a sphere
Latitudes and longitudes, called geodetic coordinates, are used for defining points in
geodesy too, but a slight deviation from the spherical assumption of earth is used
(Chapter 5)
On earth, the equator is used as the primary great circle and meridians are used as
secondary great circles
Latitude is the angle subtended at the center by the meridian arc between the equator and
the point
North or positive latitudes are used for point is on the northern hemisphere, and south or
negative latitudes for points on the southern hemisphere
The angular value of the latitude is always between 0 and 90, and is generally denoted
by ‘’
Longitude is the angle subtended by the arc on the equator between the zero or reference
meridian and the meridian through the point
Reference meridian on earth is the meridian of Greenwich
Longitude is generally denoted by  and is positive for points east of Greenwich meridian
The longitude of a point can be expressed as a value either between 0 and 360 or
between 0 and 180 east or west (of Greenwich)
Note that the latitude and longitude define only the horizontal position of a point in terms of
north or south and east or west coordinates
Spherical Coordinates and Practical Astronomy
Practical astronomy deals with the use of celestial objects in the universe to determine
positions and directions on earth such as those required in navigation and surveying
Practical astronomy is also used in precise time keeping, monitoring polar motion and
variations in earth rotation
Above are accomplished through a combination of observations and computations
Most computations involve the use of spherical coordinates to define the positions of
celestial object and solution of spherical triangles
Celestial objects in the universe include stars, sun, planets and their moons (satellites),
comets and meteors
Stars are objects emitting their own illumination and are at great distances from earth
Sun is a star and along with planets revolving around it makes the solar system
Planetary motion is governed by the Newton’s law of gravitation, and follows Kepler’s laws
Kepler’s Laws
1. The orbit of each planet is an ellipse with the sun at one focus
When the planet is farthest from the sun, it is said to be at aphelion; when it is closest to
the sun, it is at perihelion
2. As the planet revolves around the sun its radius vector sweeps out equal areas in equal
times
3. The square of the time of revolution in its orbit by each planet is proportional to the third
power of its mean distance from the sun, i.e. semi-major axis of the ellipse
Orbit of Earth
 ecliptic
 equinoxes
 solstices
Ecliptic is the orbit of the earth around the sun
Equinoxes are the points of intersection, with the ecliptic, of a line through the center of the
sun parallel to the line of intersection of the planes of the earth’s equator and the ecliptic
Solstices are points of intersection of a line, perpendicular to the line of equinoxes, with the
ecliptic
Celestial Sphere
 Definition
 Orientation
 Points and circles on the celestial sphere
Celestial sphere is a large, imaginary sphere having the same center as the center of earth
For the purpose of astronomical calculations, all celestial objects are assumed to be fixed
on the surface of the celestial sphere
Entire celestial sphere appears to be in a constant rotation around the axis of earth in a
westerly direction
The concept of celestial sphere enables us to define positions of celestial objects in space
Positions of objects on the celestial sphere can be defined using spherical coordinates
discussed earlier
This means that only the directions, and not the distances, of these objects as seen from
the center of earth are important for astronomical calculations
Important points on C. sphere
 Zenith
 North, South, East and West points
 Celestial Poles
 Vernal Equinox, Autumnal Equinox, Summer Solstice, Winter Solstice
Important Great Circles on C. sphere






Observer's meridian
C. Horizon
C. Equator
Meridians (Hour circles)
Vertical circles
Ecliptic (apparent annual path of sun)
Astronomical Coordinates
 Horizon system
 Equatorial system - I
 Equatorial system - II
These are spherical coordinates as discussed earlier and the following are fundamental to
every system
 the circle on which it is measured
 the initial point on that circle
 the direction of the measurement
 the terminal point
Horizon System
 Azimuth
(horizon, North point )
 Altitude
(vertical circle, horizon)
Azimuth (A) - Angle measured at the center (O) of the celestial sphere, on the plane of
horizon, from the north point to the foot of the vertical circle through the body (S’)
The sense of the azimuth angle is towards or through East point that is clockwise when
viewed from zenith
Azimuth of a star can have values between 0 and 360 same as azimuth of a survey line
Altitude(h) - Angle measured upward (towards zenith) at the center (O) of the celestial
sphere, on the plane of the vertical circle through the body, from the horizon to the body
Since altitude is measured from horizon toward zenith, it can have values only between 0
and 90
Altitude of a star below the horizon can be considered negative but has no useful purpose
in practical astronomy as it cannot be observed
The coordinates of a body in the horizon system are not constant, that is, mainly due to the
apparent diurnal motion of all celestial objects, their altitude and azimuth continually
change
The horizon system is local, that is, the altitude and azimuth of a star at a given instant are
different for two observers situated at different places
Just as soon as an observer changes his/her position, his/her zenith changes; hence
his/her horizon changes as well
Any change in the longitude of the observer’s position, that is, any change in the east or
west direction, the observer’s meridian too changes
For above reasons, the coordinates in the horizon system cannot be used universally
Equatorial system - I
 Hour angle
(equator, - point)
 Declination
(meridian thru star, equator)
Local hour angle (LHA) - Angle measured on the plane of the equator from  point to the
foot of the hour circle through the body
The sense of LHA is westerly, same as daily movement of star, that is a clockwise
movement when viewed from North Celestial Pole
LHA can have values between 0 and 360
Declination() - The angle measured on the hour circle (meridian) through the body from
the equator to the body
Declination can be measured toward north (celestial pole) or south, and therefore, can only
have a value between 0 and 90
A stars on the northern celestial sphere is said to have a north declinations and those on
the southern celestial sphere have south declinations
Stars have their daily movement along small circles parallel to the equator and have a
constant declination except for minute variations due to precession, nutation, polar motion,
etc.
A star having a 0 declination has its daily path along the equator and is called an
equatorial star
Declination of a star is independent of the observer’s location but LHA is not, and therefore,
they too are not suitable as universal coordinates
To an observer in the northern hemisphere (of earth), all stars having north declinations are
visible but only some south stars are visible depending on the location of the observer
The scenario is reversed for observers on the southern hemisphere
Equatorial system - II
 Right ascension
(equator, V.E.)
 Declination
(meridian, equator)
Right Ascension (R.A. or ) - Angle measured eastward on the plane of the equator, from
the Vernal equinox to the foot of the hour circle through the body
Declination – As defined earlier
Right Ascension is independent of the observer’s location as it is defined in relation to the
celestial equator and Vernal equinox
It is also independent of the time of the day as both the star and Vernal equinox move at
the same rate daily keeping their angular separation constant
For this reason the Right Ascension and Declination are used as universal coordinates to
document star positions in star catalogs that could be used by anyone anywhere in the
world
Position of the Vernal equinox too changes slightly due to effects stated earlier, and
therefore, both these coordinates need to be updated to the exact time of observations, if
highly precise results are expected
More on Local Hour Angle
Since Local Hour Angle is defined with respect to the local meridian, LHA of a star at a
given instant is different for different longitudes
This relationship can be expressed as
LHA1 - LHA2 = 2 - 1
where 1 and 2 are longitudes of two locations on earth and 1 < 2 , that is, 2 is more
easterly that 1
If both locations are in the western hemisphere, and since in practice west longitudes
increase westerly, then 1 is more easterly than 2
Above equation can be written for two locations in the western hemisphere as
LHA2 - LHA1 = 2 - 1
If Greenwich meridian is used for 1and since Greenwich meridian has zero longitude,
above two equations can be written as
GHA - LHA = w - 0 = w
LHA - GHA = e - 0 = e
w = west longitude
e = east longitude
Other Definitions
 co-latitude = 90 - 
 co-altitude = zenith distance (angle) = 90 - h
 co-declination = polar distance(p) = 90 - 
 azimuth angle () – angle between 0 and 180, either eastward or westward from North
point
 meridian angle (t) – angle between 0 and 180, either clockwise or counter-clockwise
from  point
Astronomical Triangle
A triangle on the celestial sphere formed by great circle arcs connecting celestial pole,
zenith and the celestial body
These are the intersecting great circle arcs of the observer’s meridian (PZ), and the
meridian (PS) and vertical circle (ZS) through the body
Many problems in astronomy involve solving the astronomical triangle
Examples
Concept of Time
 Epoch of time
 Interval of time
Time Systems
 Sidereal time systems based on the diurnal rotation of earth on its axis
 Ephemeris time based on planetary motion around the sun
 Atomic time based on the atomic oscillations
Sidereal Time Systems
 Sidereal Time
 Apparent Solar Time
 Mean Solar Time
Sidereal Time
The interval of time between two successive upper transits of the vernal equinox over the
same meridian is defined as the Sidereal Day
Sidereal Time at any instant, therefore, is the hour angle of the Vernal equinox
That is, Local Sidereal Time given by
LST = LHA of Vernal equinox
Since LHA of V.E. is not directly observable, LST can be determined directly by
LST = R.A. + LHA of any star
Apparent Solar Time
Interval of time between two successive lower transits of the sun’s center over the same
meridian is the Apparent Solar Day
Local Apparent Time, therefore, is reckoned from the lower transit of apparent sun and,
there is a difference of 12 hours between the LHA of apparent sun and local apparent time
That is
LAT = LHA (of apparent sun) + 12h
Mean Solar Time
The interval of time between two successive lower transits of the mean sun over the same
meridian is Mean Solar Day
Again, Mean Solar Time and LHA of mean sun are 12 hours apart, that is
LMT = LHA (of mean sun) + 12h
Other Time Standards




Standard time (Zone time)
Universal time (UT)
Coordinated Universal Time (UTC)
Broadcast Time
Realize that all three time standards defined earlier are defined using local hour angle of
celestial bodies, and therefore, are referenced to the local meridian
This means that, at any given instant, locations with different longitudes have different
times
This is not suitable for civil time keeping as it is very inconvenient in any operation
Standard Time or Zone Time has been defined to overcome this inconvenience by having
the same time over a larger geographic region
Standard Time for a certain region is generally defined to be an integral number of hours
ahead or behind GMT (UTC)
Standard Times used by countries in the eastern hemisphere are ahead of GMT and those
used by countries in the western hemisphere are behind GMT
Standard time of a region is also equal to the LMT referenced to a meridian, called
standard meridian, within that region
The longitude value of the standard meridian is a multiple of 15, in general, e.g. Eastern
Standard Time (EST) is equivalent to LMT at longitude 75 W
Some countries in high latitudes such as USA and Canada, advance their clocks in the
spring to begin the day early, is called Daylight Saving Time, or in some European
countries, Summer Time
Universal Time
Universal Time (UT) refers to GMT, but there are several versions of Universal Time
UT0 - GMT as determined by observations
UT1 - UT0 after being corrected for polar motion
UT2 - UT1 corrected for irregularities of earth's rotational speed
Coordinated Universal Time (UTC) is a time used internationally that is derived from both
UT1 and International Atomic Time (TAI)
Since UT1 is not quite uniform, it has a constant deviation from TAI
This deviation is corrected from time to time by adding one second called leap second
This is a step adjustment carried out when necessary either on June 30th or December 31st
or on both days
This ensures that the difference between UT1 and UTC, called DUT1, is less than 0.9s
Broadcast time such as that by WWV at Fort Collins, CO, broadcast UTC along with a
coded DUT1 correction
Conversion of Time

Standard time to GMT

Mean time to apparent time and vice versa

Mean time to sidereal time and vice versa
Between any two time standards, there is a difference in the initial epoch as well as a
difference in time scale
Both these have to be known in order to convert one time standard to another
GMT from Standard Time
This conversion is straightforward and is done by adding/subtracting an integral number of
hours to/from GMT
For example,
Or
GMT = EST + 5h
GMT = EDT + 4h
Conversion of Mean Time to Apparent Time
The difference between Mean time and Apparent time at any instant is called Equation of
Time (E), i.e.
E = AT - MT
Equation of time includes both the epoch difference and scale difference
Equation of time may be generated from available mathematical expressions
E is usually tabulated in most Solar ephemeris for 0h GMT for everyday of the year
A value of ‘E’ at any other time may be found by a simple (linear) interpolation with GMT of
the instant as the independent variable, even though the function itself is not linear
Conversion of Mean time to Sidereal time and vice versa
ST = RA + HA
ST = RA(ms) + HA(ms)
ST = RA(ms) + MT - 12h
GST = RA(ms) + GMT - 12h
GST at 0h GMT = RA(ms) - 12h
In practice, this value is given, for everyday of the year, in star catalogs or nautical
almanacs
This means that conversion of sidereal time to mean time or vice versa in practice boils
down to a conversion of elapsed time interval since 0h GMT
There is a linear relationship between mean and sidereal time intervals
This relationship is found by using the number of mean days and sidereal days in a
Tropical Year
Tropical Year is the time interval between two successive passages of the sun through the
vernal equinox
It has been found that
1 Tropical year = 365.2422 Mean days = 366.2422 Sidereal days
That is,
1 Mean Day = 366.2422 = 1.00273791 Sidereal Days
365.2422
or
1 Sidereal Day = 365.2422 = 0.99726957 Mean Days
366.2422
Above relationship holds true for hours, minutes, and seconds as well and can be used for
conversion of time intervals only
As stated earlier, conversion of an epoch of time between Mean time and Sidereal time
involves two steps
 Finding both MT and ST at 0h GMT
 Converting elapsed time from one to the other
Note that the Mean day is longer than the Sidereal day, and therefore, one unit of mean
time is larger (longer) than one unit of sidereal time
This means that fewer mean time units than sidereal time units are needed to express the
same time interval and vice versa
Conversion of time is needed in some astronomical observations and computations
Direction Determination
Directions on earth are important in navigation, surveying, etc.
Directions are indicated as azimuths referenced to true (geographic) north
The meridian at any point on earth runs exactly north-south
Azimuth of a line on the earth surface is the angle, reckoned clockwise, between the north
meridian and the line
Recall that the azimuth of a celestial body is defined as the angle between the observer's
meridian and the vertical circle through the celestial body, reckoned easterly from the north
side of the observer's meridian
This is also equal to the clockwise angle between the observer's meridian and the vertical
circle, as projected on the plane of the horizon, from north
The azimuth of a celestial body can be computed very easily at any instant, given the
latitude of the location
If the angle between the celestial body and the line on earth is measured at the same
instant, the azimuth of the line on earth can be deduced
There are two basic methods used in practice to compute azimuth of a celestial body
Altitude method - By measuring the altitude of the celestial body at a given instant
Hour Angle method - By using the local hour angle of the body deduced from time
measurements
Altitude method is less accurate than the hour angle method, in general
Stars offer a better accuracy than sun, in general, and are used in high precision work
Advantage of the altitude method is that the altitude of a celestial body is directly
measurable and the time of observation need not be known very precisely
In hour angle method, the time needs to be recorded to a fraction of a second except for
circumpolar stars
Instruments used in these observations must be consistent with the accuracy sought
Errors in Astronomic Observations
Observed data to be used in computations must be corrected for various errors inherent in
observations
These errors may be due to imperfections in the instruments, physical causes or even due
to the carelessness or inexperience of the observer
The measurement errors usually fall into three categories, namely, blunders or gross
errors, systematic errors, and random errors
There are also errors arising from the fact that the measurements made to the astronomical
body do not fit into the mathematical model used for computations
Yet another source of error is the position of the astronomical body in the sky at the time of
observations because the computations are sensitive to certain configurations of the
astronomical triangle
Only three errors considered at this time, namely, Atmospheric refraction, parallax, and
semi-diameter correction of the sun, all of which are systematic errors
Atmospheric Refraction
This is the continuous and systematic bending of light rays as they travel through the
atmosphere, and is sometimes called astronomic refraction or simply refraction
This is caused by the varying density of atmosphere
Refraction results in change of altitude of celestial objects as measured, i.e. the
apparent altitude is different from the true altitude
For this reason, the measured altitudes (or zenith angles) of all astronomical bodies must
be corrected before they are used in any calculations
Magnitude of the correction depends on the altitude of the body, and the temperature,
pressure, and humidity of the atmosphere
The refraction is larger at lower altitudes and the computed value is never more than 3
minutes of arc in most practical situations
The effect of refraction is to make the observed altitude larger than true altitude or make
the measured zenith angle smaller than true zenith angle
Consequently, the computed correction is subtracted from measured altitude and added to
measured zenith angle
Parallax
This is a correction necessary to bring the point of observation to the center of the earth
For all theoretical considerations, the observer's position was considered to be same as the
center of the earth
This is perfectly all right for objects as far away as stars since the difference in angular
measurements to these objects introduced by the radius of the earth is negligible
But when making observations to the sun, this difference cannot be disregarded due to its
proximity to earth
Parallax has no effect on the azimuth of a celestial object, as the horizon defined at the
point of observation is parallel to the horizon through the center of earth
Therefore, this correction needs to be applied only to measured altitudes or zenith angles
of sun
The magnitude of the correction depends on the altitude, is less than 8.8 arc seconds, and
is added to the measured altitude or subtracted from measured zenith angle
Both refraction and parallax corrections are small and they can be applied to the measured
zenith angle or altitude in any order
Semi-Diameter (of Sun)
In theoretical considerations, the sun is assumed to be a point object in the sky
It is not generally possible to observe the sun’s center directly unless special optics are
used
The observations are generally made to the edges, called limbs, of the sun and a correction
called semi-diameter correction is applied
The angular value of the semi-diameter can be found in solar ephemeris and varies
throughout the year as the distance between the earth and sun changes
This is the angle subtended at the center of the earth by the sun’s semi-diameter (radius),
and is known as the angular (geocentric) semi-diameter of the sun
The correction to the vertical angle (if measured) is exactly equal to the angular value of the
semi-diameter of sun
This could also be considered as the angle subtended at the instrument by the radius of the
sun and should be added or subtracted from the measured vertical angle depending on
whether the vertical angle is measured to the lower or upper edge of the sun
Semi-diameter correction applied to the horizontal angle between the sun and the survey
line also depends on whether the angle is measured to the left or right edge of the sun
In addition, it also varies with the altitude of the sun at the time of measurement
This is because the angle subtended at the instrument by the horizontal radius of sun is in
a plane inclined to the horizon
The horizontal angle between two vertical circles passing through the ends of this
horizontal radius is larger than the angle on the inclined plane
In other words, the separation between two vertical circles passing through ends of sun’s
semi-diameter is larger at the horizon than at the altitude of sun
Therefore, the semi-diameter correction applied to the horizontal angle is computed from
SH = SD/cos h = SD/sin z
SD = semi-diameter value for the date of observations from solar ephemeris
h/z = vertical/zenith angle to the sun
Note that the vertical or the zenith angle need not be measured but can be computed if the
time of observation has been recorded, e.g. Hour angle method of sun shots
The computed correction should be added to the clockwise horizontal angle measured from
the survey line to the sun, if left edge of the sun has been sighted or subtracted from the
clockwise horizontal angle if right edge has been sighted