Introduction In a clear night, the sky appears as a vast hollow hemisphere with its interior studded with innumerable stars. On observing the sky for some duration it appears that the celestial bodies are revolving around the earth with its centre at the position of the observer. The stars move in a regular manner and maintain same position relative to each other. Consequently, terrestrial position or direction defined with reference to celestial body remains absolute for all practical purposes in plane surveying. Thus, the absolute direction of a line can be determined from the position / direction of a celestial body. Astronomical Terms and Definitions To observe the positions / direction and movement of the celestial bodies, an imaginary sphere of infinite radius is conceptualized having its centre at the centre of the earth. The stars are studded over the inner surface of the sphere and the earth is represented as a point at the centre. Figure 25.1 shows a celestial sphere and some principal parameters necessary to understand astronomical observation and calculations for determination of absolute direction of a line. The important terms and definitions related to field astronomy are as follows:
Transcript
Introduction
In a clear night, the sky appears as a vast hollow hemisphere with its interior studded with innumerable stars. On observing the sky for some duration it appears that the celestial bodies are revolving around the earth with its centre at the position of the observer. The stars move in a regular manner and maintain same position relative to each other. Consequently, terrestrial position or direction defined with reference to celestial body remains absolute for all practical purposes in plane surveying. Thus, the absolute direction of a line can be determined from the position / direction of a celestial body.
Astronomical Terms and Definitions
To observe the positions / direction and movement of the celestial bodies, an imaginary sphere of infinite radius is conceptualized having its centre at the centre of the earth. The stars are studded over the inner surface of the sphere and the earth is represented as a point at the centre. Figure 25.1 shows a celestial sphere and some principal parameters necessary to understand astronomical observation and calculations for determination of absolute direction of a line. The important terms and definitions related to field astronomy are as follows:
Celestial sphere : An imaginary sphere of infinite radius with the earth at its centre and other celestial bodies studded on its inside surface is known as celestial sphere.
Great Circle (G.C) : The imaginary line of intersection of an infinite plane, passing through the centre of the earth and the circumference of the celestial sphere is known as great circle.
Zenith (Z) : If a plumb line through an observer is extended upward, the imaginary point at which it appears to intersect the celestial sphere is known as Zenith. The imaginary point at which it appears to intersect downward in the celestial sphere is known as Nadir (N).
Vertical circle : Great circle passing through zenith and nadir is known as vertical circle.
Horizon: Great circle perpendicular to the line joining the Zenith and Nadir is known as horizon.
Poles : If the axis of rotation of the earth is imagined to be extended infinitely in both directions, the points at which it meets the celestial sphere are known as poles. The point of intersection in the northern hemisphere is known as north celestial pole and that in the southern hemisphere as south celestial pole.
Equator : The line of intersection of an infinite plane passing through the centre of the earth and perpendicular to the line joining celestial poles with the celestial sphere.
Hour circle : Great circle passing through celestial poles is known as hour circle, also known as declination circle.
Meridian : The hour circle passing through observer's zenith and nadir is known as (observer's) meridian. It represents the North-South direction at observer station.
Altitude (h) : The altitude of a celestial body is the angular distance measured along a vertical circle passing through the body. It is considered positive if the angle measured is above horizon and below horizon, it is considered as negative.
Azimuth (A) : The azimuth of a celestial body is the angular distance measured along the horizon from the observer's meridian to the foot of the vertical circle passing through the celestial body (Figure 25.2).
Declination () : The declination of a celestial body is the angular distance measured
from the equator to the celestial body along the arc of an hour circle. It is considered positive in North direction and negative in South.
Ecliptic : The great circle along which the sun appears to move round the earth in a year is called the ecliptic.
Equinoctial points : The points of intersection of the ecliptic circle with the equatorial circle are known as equinoctial points. The point at which the sun transits from Southern
to Northern hemisphere is known as First point of Aeries () and from Northern to
Southern hemisphere as First point of Libra ().
Right ascension : The right ascension of a celestial body is the angular distance along
the arc of celestial equator measured from the First point of Aeries () to the foot of the
hour circle. It is measured from East to West direction i.e., anti-clockwise in Northern hemisphere.
Prime meridian : Reference meridian that passes through the Royal Naval Observatory in Greenwich, England is known as prime meridian; it is also known as Greenwich meridian.
Longitude () : The longitude of an observer's station is the angular distance measured
along the equator from the prime meridian to the observer's meridian. It varies from zero degrees to 180° E and 0° to 180° W.
Latitude (): The latitude of an observer's station is the angular distance measured
along the observer's meridian from the equator to the zenith point. It varies from zero degree to 90° N and 0° to 90° S
Hour angle (HA) : The hour angle of a celestial body is the angle at the equatorial plane measured westward from meridian to the hour circle passing through the celestial body (Figure 25.3).
Local hour angle (LHA): The angular distance of a celestial body measured westward from the point of intersection of the equator and the meridian of the observer to the foot of the hour circle passing through the celestial body.
Greenwich hour angle (GHA) : Angle at the equatorial plane measured westward from the prime (Greenwich) meridian to the hour circle through the celestial body.
Spherical triangle: Triangle formed by the intersection of three arcs of great circles (on the surface of the celestial sphere) is known as spherical triangle.
The dimension of the celestial sphere is so large that the position of the observer and the centre of the earth appears to be the same point
Astronomical Triangle
The spherical triangle formed by arcs of observer's meridian, vertical circle as well as hour circle through the same celestial body is known as an astronomical triangle. The vertices of an astronomical triangle are Zenith point (Z), celestial pole (P) and the celestial body (S) and thus termed as ZPS triangle (Figure 25.4a). In each astronomical triangle, there are six important elements. Three of them are the three sides and other three are the three angles of the triangle. It is important to know these elements as some of these will be required to be observed in the field and others are to be computed to find the position / direction of celestial body.
Polar distance (PS or p) : The angular distance from the celestial pole (P) to the celestial body (S) along the hour circle is known as polar distance. It is also known as
co-declination and is designated by p (= 90°- ), where is the declination of the
celestial body, S.
Zenith distance (ZS or z) : The angular distance from observer's zenith (Z) to the celestial body (S) along the vertical circle is known as zenith distance. It is also known
as co-altitude and is designated by z (= 90°- h), where h is the altitude of the celestial body, S.
Co-latitude, ZP : The angular distance from observer's zenith (Z) to the celestial pole
(P) along the observer's meridian is known as co-latitude and is given by (90°- ),
where is the latitude of the observer
Angle Z : The angle at the zenith (A) is measured from the observer's meridian to the vertical circle passing through the celestial body in a plane parallel to the observer's horizon. It is nothing but the azimuth of the celestial body (Figure 25.2). It is measured clockwise from the observer's meridian and its value ranges from zero to 360°.
Angle P : The angle at the pole (P) is measured from the observer's meridian to the hour circle passing through the celestial body in a plane parallel to the equatorial plane. It is nothing but (360°– H, hour angle of the celestial body) (Figure 25.4b). Hour angle is measured clockwise from the upper branch of the observer's meridian.
Angle S : angle at a celestial body between the hour circle and the vertical circle passing through the celestial body. It is known as the parallactic angle.
If any of the three elements are known, the remaining three can be computed from formulae of spherical trigonometry (Appendix- 25A)
Example
Ex25-1 Find the shortest distance between a station (29° 52' N, 77° 54' E) at Roorkee and to a station (28° 34' N, 77° 06' E) at Delhi. Determine the azimuth of the line along which the direction of the shortest distance to be set out starting from Roorkee.
The shortest distance between two stations on the surface of the earth lies along the circumfrence of the great circle passing through the stations.
Refering Figure Example 25-1, let us consider a great arc RD passing through the Roorkee and Delhi respectively. Thus, arc RD is the shortest distance between the stations. Let P be the pole of the earth. and PD and PR are arcs of meredians passing through Delhi and Roorkee stations respectively.
Then, PDR is a astronomical triangle, where
P = 77° 54' - 77° 06' = 48'
Distance PD, r = 90° - 28° 34' = 61° 26
Distance PR, d = 90° - 29° 52' = 60° 08'
Using equation (25A.3a),
Or, cos p = cos d . cos r + cos P . sin d . sin r
= cos 60° 08' . cos 61° 26' + cos 48' . sin 60° 08' . sin 61° 26'
= 0.23826 + 0.76154 = 0.99966837
Thereforep1°28' 33".48
Assuming, rdius of the earth, R = 6370 km.
Arc distance, RD =
For the determination of the direction from R to D, the angle R of the spherical triangle is required to be determined.
Using equation 25A.11a,
Therefore R + D = 179° 36' 34".12 ------------------------Equation 1
Again, using equation 25A.11b
= 3° 43' 24".07
Or, R - D = 123° 31' 05".20 ------------------------Equation 2
Solving Equation 1 & 2,
R = 151° 33' 49".66
Thus, the azimuth of the line to be set out at station R to proceed along shortest path to the station at Delhi is = 360° - R = 208° 26' 10".34
Azimuth of a Line
Azimuth of a line is its horizontal angle measured clockwise from geographic or true meridian.
For field observation, the most stable and retraceable reference is geographic north. Geographic north is based on the direction of gravity (vertical) and axis of rotation of the earth. A direction determined from celestial observations results in astronomic (Geographic) north reference meridian and is known as geographic or true meridian.
The azimuth of a line is determined from the azimuth of a celestial body. For this, the horizontal angle between the line and the line of sight to the celestial body is required to be observed during astronomic observation along with other celestial observation.
Let AB be the line whose azimuth (AAB) is required to be determined (Figure 26.1). Let O be the station for celestial observations. Let S be the celestial body whose azimuth (As) is determined from the astronomical observation taken at O. The horizontal angle from the line AB to the line of sight to celestial body (at the station O) is observed to
be ° clockwise. The azimuth of the line, AB can be computed from
AAB = AS - ° (clockwise).
If AAB computes to be negative, 360° is added to normalize the azimuth
In order to compute the azimuth of a line with proper sign, it is better to draw the known
parameters. The diagram itself provides the azimuth of the line with proper sign. For example,
in Figure 26.2, first a line of sight to celestial body, OS is drawn. Then, the azimuth of the
celestial body, AS is considered in counter-clockwise from the line OS and obtained the true
north direction i.e, the line ON. Similarly, the horizontal angle ° is represented in counter
clockwise (since the angle from the line to the celestial body is measured clockwise) direction
from OS to obtain the relative position of the line. The angle NOB represents the azimuth of the
line AB.
Determination of Azimuth of Celestial body
In field astronomy, a celestial body provides the reference direction. So, from the geographic location (latitude and longitude) of the station, ephemeris data of celestial body and either time or altitude of the same celestial body, the azimuth of the celestial body is computed by solving astronomical triangle. If time is used, the procedure is known as the hour-angle method. Likewise, if altitude is measured, the procedure is termed as the altitude method. The basic difference between these two methods is that
the altitude method requires observation of approximate time and an accurate vertical angle of the celestial body, whereas the hour angle method requires observation of accurate time.
Recent developments of time receivers and accurate timepieces, particularly digital watches with split-time features, and time modules for calculators, the hour-angle method is more accurate, faster. It requires shorter training for proficiency. It has fewer restrictions on time of day and geographic location and thus is more versatile. The method is applicable to the sun, Polaris, and other stars. Consequently, the hour-angle method is emphasized, and its use by surveyors is encouraged.
Hour Angle Method
In this method, precise time is being noted when the considered celestial body is being bisected. The observed time is used to derive the hour angle and declination of the celestial body at the instant of observation.
The geographic position (latitude and longitude) of the observation station is required to be known a priori for the hour angle method. Usually, these values are readily obtained from available maps. However, to achieve better accuracy, latitude and longitude must be more accurately determined specially during observations for celestial bodies close to the equator--e.g., the sun-than for bodies near the pole--e.g., Polaris
The declination of the celestial bodies at the instant of observation is required to be known for computation of azimuth of the celestial body. It is available in star almanac at the 0, 6, 12 and 18 hours of UTI of each day ( Greenwich date). Thus, the declination at the instant of observation (of celestial body) is determined by linear interpolation for corresponding the UT1 time of observation. However, since the declination of the sun
varies rapidly, its interpolation is done using the relation:
Declination, d = Decl 0h + (Decl 24h - Decl 0h) ( ) + (0.0000395) (Decl 0h) sin (7.5
UT1) -------------(Equation 26.2)
The hour-angle of the celestial body is being derived using the GHA (available in star almanac with reference to Greenwich date) and the longitude of the observation station. For observations in the Western Hemisphere, if UTI is greater than local time, the Greenwich date is the same as local date and if UTI is less than local time, Greenwich date is the local date plus one day. For the Eastern Hemisphere, if UTI is less than local time (24-hr basis), Greenwich date is the same as local date and if UTI is greater than local time, Greenwich date is local date minus one day. The hour angle of the celestial body at the observation station is the LHA. Thus, it is the LHA at UTI time of observation which is necessary to compute the azimuth of a celestial body. Hence, as can be seen from Figure 26.3, the equation for the LHA is
LHA = GHA - W (west longitude) ----------------- (Equation 26.3)
Or LHA = GHA + E (east longitude) -------------------(Equation 26.4)
LHA should be normalized to between 0° and 360° by adding or subtracting 360°, if necessary.
The Greenwich hour angle (GHA) of celestial bodies-the sun, Polaris, and selected stars-is tabulated in star almanac from 0 hr to 24 hr at an interval of 6 hours of UTI time of each day (Greenwich date). Thus, to find GHA at the time of observation linear interpolation is required to be performed.
The GHA can also be derived by making use of the equation of time E (apparent time minus mean time) by using the relation:
GHA = 180° + 15 E ---------------------------Equation (26.5)
where E is in decimal hours. In those cases where E is listed as mean time minus apparent time, the algebraic sign of E should be reversed.
Once the parameters (declination and Hour angle of the celestial body, latitude of the observation station) required to compute the azimuth of the celestial body are available, the computation of azimuth of the celestial body is carried out using the relations of astronomical triangle (Appendix- 25A)
Figure 25A shows a spherical triangle ZSP. In all there are six quantities in a spherical triangle, namely, three angles Z, S and P and three sides z, s and p. If any three quantities are known, the remaining three can be computed from different formulae of the spherical trignometry given below.
Equation 25A.5 to 25A.7 should be used when there is some confusion about the signs of the
angles. Because the value of the angle A / 2 is always less than 90°, there is no ambiguity. As
far as possible, Eq. 25A.5 should be used when A < 90°. Eq. 25A.7 can be used for any value of
the angle.
Examples
Ex26-1 A celestial body is observed at a station (latitude 36° 04' 00" N and longitude 94° 10' 08" N). The UTI at the instant of observation was (15h 16m 41s.57).The GHA at 0hr (UT1) was found to be 177° 04 ' 44".1 on the day of observation and 177° 08' 06".3
on the next day. The declination of the selected body at 0hr (UT1) was + 6° 15' 05".9 S and + 5° 5' 54".7 S on the day of observation and next day respectively. Determine the azimuth of the celestial body.
Soluton : Given, GHA at 0hr = 177° 04 ' 44".1
GHA at 24hr = 177° 08' 06".3
Instant of observation of the celestial body = 15h 16m 41s.57
Now, using the relation (Equation 26.3), azimuth of the celestial body
Z =
=
= - 57.10487745°
Since, LHA is between 180° and 360° and A is negative (Figure Ex 26.1), thus
Azimuth of the body, A = Z + 180° = - 57.10487745° + 180°
= 122.8951226° = 122° 53' 42 ".44
Ex25-2 Determine the azimuth of the sun having declination 8° 24' S during sunset at Roorkee (latitude 29° 52' N).
Solution :
Given, latitude of observer, = 29° 52'
declinationof the sun, = 8° 24' S
The Sun at sunset, thus altitude = 0°
Therefore = 90°
z = 90° += 98° 24'
s = 90° - = 60° 08'
Figure Example 25-2 (b)
The triangle ZPS is a right angled astronomical triangle, where RZ (in ZPS) is in western hemisphere.
Now, using equation 25A.3a, we get
= - 0.168
Z = 99° 41' 53".07 (W)
Thus, azimuth of the sun, A = 360° - Z = 261° 59' 32".26
Sun Observations (Hour- Angle Method)
Among all the celestial bodies, the sun is the most prominent body that can be observed easily and accurately. Thus, the sun observation provides surveyors a convenient method for determination of astronomic azimuth. In the hour-angle method, a horizontal angle from a line to the sun is measured. Knowing accurate time of the observation and geographic position of the observation station, the sun's azimuth is computed using the relations of astronomical triangle. This azimuth and horizontal angle are combined to yield the line's azimuth as explained in Lesson 26.
The sun is observed through the telescope fitted with either an eyepiece sun glass or an objective lens filter. For total stations, an objective lens filter is mandatory (to protect EDMI components). It is to be noted that the sun's image is large in diameter-approximately 32 min of arc-making accurate pointing on the center impractical. Thus, in lieu of pointing the centre, observation to the sun are usually taken with the edges tangential to both the horizontal and vertical cross hairs. It is usually achieved by allowing sun's trailing edge to move onto the vertical cross hair or the leading edge is pointed by moving the vertical cross hair forward, until it becomes tangent to the sun's image.
The field work involved in the determination of azimuth of a line from sun observation consists of following steps:
1. Carry out temporary adjustment of a theodolite at the observation station with face left condition.
2. Open the lower plate main screw and swing the telescope to bisect the reference object. Fix the lower plate main screw and bisect accurately using the lower plate tangent screw. Note down the horizontal circle reading.
3. Swing the telescope by opening the upper plate main screw. Bring the image of the sun into the upper left quadrant of the diaphragm (Figure 27.1a). Close the upper plate main screw and then bring the vertical hair tangent to right limb of the sun using the upper plate tangent screw. As soon as the lower limb of the sun makes contact with the horizontal hair, the chronometer time is recorded. Note down the horizontal (as well as vertical, if altitude of the sun is required to be observed) circle readings.
4. Next, bring the image of the sun into lower right quadrant using the upper and vertical plate tangent screws. The vertical hair is kept in tangent to left limb of the sun using by the upper plate tangent screw. As soon as the upper limb of the sun makes contact with the horizontal hair, the chronometer time is recorded (Figure 27.1b). Note down the horizontal (as well as vertical, if altitude of the sun is required to be observed) circle readings.
5. Open the upper plate main screw and swing back the telescope to the reference object. Close the upper plate main screw and bisect the reference object. Note down the horizontal circle reading.
6. Change the face left of the instrument into the face right condition and repeat step 2 with the instrument in face right condition.
7. Swing the telescope by opening the upper plate main screw. Bring the image of the sun into the upper right quadrant of the diaphragm (Figure 27.1c). Close the upper plate main screw and then bring the vertical hair tangent to left limb of the sun using the upper plate tangent screw. As soon as the lower limb of the sun makes contact with the horizontal hair, the chronometer time is recorded. Note down the horizontal (as well as vertical, if altitude of the sun is required to be observed) circle readings.
8. Next, bring the image of the sun into lower left quadrant using the upper and vertical plate tangent screws. The vertical hair is kept in tangent to right limb of the sun using the upper plate tangent screw. As soon as the upper limb of the sun makes contact with the horizontal hair, the chronometer time is recorded (Figure 27.1d). Note down the horizontal (as well as vertical, if altitude of the sun is required to be observed) circle readings.
Thus, a set of observation is to be taken. It consists of four instants of time and four horizontal circle readings (and four vertical circle readings). An azimuth of the line AZL should be computed for each pointing on the sun.
For any field observation, it may consist of one or more sets, but a minimum of three sets is recommended.
Errors and Correction
The errors involved in determining a line's azimuth can be divided into two categories: (I) measuring horizontal angles from a line to the sun and (2) errors in determining the sun's azimuth. Except when pointing on the sun, errors in horizontal angles are similar to any other field angle.
Since, the observations are taken at the edges of the sun, a correction for semi-diameter is required. The correction (C) to be applied to the measured horizontal angles is:
where is the (correct) altitude of the sun and is the semi-diameter of the sun (can
be obtained from star almanac).
The width of a theodolite cross hair is approximately 2 or 3 arc-sec. With practice, the sun's edge, particularly the trailing one, can be pointed to within this width. In many instances, pointing the sun may introduce a smaller error than pointing the backsight mark.
Total error in the sun's azimuth is a function of errors in obtaining UTI time, and scaling latitude and longitude. The magnitude these errors contribute to total error which is a function of the observer's latitude, declination of the sun, and time from local noon .
Since errors in scaling latitude and longitude are constant for all data sets of an observation, each computed azimuth of the sun contains a constant error. Errors in time affect azimuth in a similar manner. Consequently, increasing the number of data sets does not appreciably reduce the sun's azimuth error. The increase can, however, improve horizontal angle accuracy and therefore have a desirable effect.
After azimuths of the line have been computed, they are compared and, if found within acceptable limits, averaged. Azimuths computed with telescope direct and telescope reversed should be compared independently. Systematic instrument errors and use of an objective lens filter can cause a significant difference between the two. An equal
number of direct and reverse azimuths should be averaged.
An alternate calculation procedure averages times and angles, or points on opposite edges of the sun and averages to eliminate semi-diameter corrections. Due to the sun traveling on an apparent curved path and its changing semi-diameter correction with altitude, this procedure usually introduces a significant error in azimuth. Also, it does not provide a good check on the final azimuth.
Determination of the altitude of the sun
The altitude of the sun can be computed from astronomical triangle (Figure 26.3) using the relation
h = sin-1 (sin sin + cos cos cos LHA) --------------------- Equation (27.1)
The altitude of the sun can also be obtained from the observation of the sun by taking the vertical angle measurement along with horizontal angle. In this case, the observations need to be updated with the following corrections:
(a) Correction for refraction: Mean value of refraction (r0) based on observed altitude is available from star almanac. A table for modification factor (f), depending on the temperature and pressure during observation, is also available in star almanac. Thus,
Correction for refraction from star almanac is Cr = ro . Other wise, can be computed
using Cr = -58" cos . It is always subtractive from the observed altitude.
(b) Correction for parallax: A correction strictly needs to be applied for the Sun's
parallax, which will suffice to be considered as 0.1 for all altitudes less than 70°. Otherwise, it may be computed using
Cp = +8.82 cos
The correction for parallax is to be added always to the observed altitude.
(c) Correction for semi-diameter
Example
Ex27-1 On September 29, 2003 the sun was observed on the rooftop of Geomatics Engineering section of I.I.T. Roorkee having geographical location (Latitude 29° 50'
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