Various Terms and Definitions used in Field Astronomy (Surveying)

Hello!

Field astronomy deals with the determination of the relative positions of the celestial bodies by taking the astronomical observations.
       

Related terms:

1. Celestial sphere: If we assume the space to be a sphere having the earth as its center and all the star lying on its surface, or studded in it. The celestial sphere can be of few kilometers to many thousand kilometer.

2. Zenith and Nadir : These are two points on the celestial sphere opposite to each other and lying above and below the observer. Zenith is the point on the celestial sphere, above the head of the observer and Zenith is the point on the celestial sphere below the observer. Alternatively, these are the points of intersection of the plumb line(drawn through the point of observation) with the celestial sphere.

3.Terrestrial poles and equator: Terrestrial poles are the points of intersection of the axis of rotation of the earth with the earth sphere, and the terrestrial equator is the great circle of the earth which is perpendicular to the the axis of rotation of the earth.

4. Celestial poles and Celestial equator: If the earth's axis of rotation is extending on both direction, it will intersect with the celestial sphere at the two points, celestial poles. Similarly Celestial equator is the great circle of the celestial sphere, in which the plane of intersection of the terrestrial equator with the celestial sphere lies.

5. Sensible Horizon: In the celestial sphere the point of observation is taken as the center. Sensible horizon is a great circle of the celestial sphere which passes through the point of observation and is tangential to the earth surface, or which is perpendicular to the zenith-nadir line.

6.Vertical Circle: The vertical circle of a celestial sphere is a circle passing through the zenith and nadir and therefore all the vertical circles are perpendicular to the horizon.

7. Observer's meridian: It is a circle which passes through the zenith and nadir of the observation point as well as through the poles of the celestial sphere. So it is a vertical circle.

8.Prime meridian: It is a vertical circle which is at right angles to the observer's meridian.

9. Azimuth: It is the angular distance between the observer's meridian and the vertical circle passing through the observer(Zenith and Nadir) and the heavenly body.

10. Hour Angle: It is the angular distance between the declination circle and the observer's meridian.

11. Latitude: It is the angular distance of the zenith from the equator.

12. Co-latitude: It is complementary angle of the latitude, i.e. 90 - latitude. It is also known as the zenith distance from the poles.

13. Right Ascension: Right ascension is the angular distance along the equator of the heavenly body from the point of Aries. It is simply written as R.A. and is always measured in the right direction from 0 to 360.

14. Ecliptic: It is the the path of the Sun around the earth assuming the earth to be stationary, traveled in one complete year. Ecliptic intersects the equator at the point of Aries and the Libra just opposite to Aries. It is the spring season when summer enters into Aries to the northern hemisphere, and it is the start of winter in the northern hemisphere when it passes the Libra and enters into the southern hemisphere.

There are some important things to discuss to understand the position of the star, or a heavenly body.

Napier's Rule: Napier's rule can be be used to solve the spherical trigonometry dealing with the right angled spherical triangles. If we arrange the 5 remaining angles in a circle in the manner as shown in fig. below, then we can get the other three angles if two of the angles are known.

The formula used is written in the photo itself. It says the sine of the middle angle is equal to the product of the tangents of the adjacent angles and product of the cosines of the opposite angles.

Star at elongation: The star is said to be at elongation when it is at its farthest point towards east or towards the west from the observer's meridian. This is point where the path of the star is tangent to the vertical circle passing through the star and the zenith nadir line. So in this position the star angle M is 90 degrees.

Star at prime vertical: When the star is on the prime vertical which is the vertical circle at right angle to the observer's meridian, the star is said to be at the prime vertical. So in this position the Azimuth is 90 degrees.

Star at horizon: When the star is at the horizon then the altitude of the star is zero, so the co-altitude or the zenith distance is 90 degrees.

Star at culmination: When the star is on the observer's meridian, either culminating from the east to west or from west to east, the star is said to be at culmination.

Aphelion: This is the point on the elliptic path of the Sun when it is at its farthest distance from the earth (Earth is assumed to be stationary, at one of the foci of the ellipse.)

Perihelion: Perihelion is another point on the ecliptic when the Sun is at the nearest distance from the earth. When the Sun is at its nearest distance to the earth, the apparent motion of the Sun is faster, as compared to other positions. 

Note:The Sun is always stationary, but it astronomy we assume the earth to be the center of the universe, so we assume the Sun to be moving. It is called its apparent motion.

Sidereal Time: It is a time which is obtained using the sidereal time system. In the Sidereal time system, the time at any place can be measured by measuring the longitude eastwards from the first point of Aries to the meridian of the place along the equator.

If at any moment, Sun's Right Ascension is known and the hour angle is known, then we can find the Local sidereal time = Right Ascension of the sun + Hour angle of the Sun

Apparent/ True Solar Time: It is the time obtained based on the Sun's motion above a given place. In this system the time is measured as 0 hours 0 min 0 Sec, when the Sun is at its lower transit, means when it is midnight at the place. It is 12 Hours 0min 0sec, when the Sun is over the head of the place. So a day is the time interval between two consecutive transits of the Sun from the place.

The day is divided into 24 hours, so each hour Sun moves 15 degrees westwards. unfortunately the length of the day is not the same throughout the year, so this system can not be adopted in the digital watches of the modern day.

The days are of variable length due to the obliquity of the path of the Sun and the non-uniform motion of the Sun around the earth.

Mean Solar Time: So a new system is originated to measure the time, it is known as the mean solar time. There is a mean Sun, which revolves around the earth in a uniform speed, on the equatorial path. So the days are of the uniform length throughout the year. A mean solar day is the average of the lengths of the 365 days of a year. The watches give us the mean solar time.

Equation of Time: So as we know that there is a difference between the true solar time and the mean solar time, this difference is known as the equation of time.

So the equation of time = Apparent Solar time - Mean solar time

The apparent solar time and mean solar time, are one behind the other at various time periods of a year. Sometimes, the Apparent solar time is forward of Mean Solar time and sometime vice verse. So the equation of time is either positive or it is negative. There are four times in a year when the equation of time becomes zero, one such day is April 16 of the year.

It can all be understood by considering the obliquity of the path of the Sun and the non-uniform motion of the Sun around the earth.

Standard Time: To have the same time at all the places on the earth, the meridian passing through the Greenwich is taken as the standard. So when the mean Sun passes through the standard meridian, the standard time is 12h 0m 0sec in the noon. The standard time is same at all the places in the earth, but the local times can be found out by knowing the longitude of the place, from the Greenwich meridian, and adding or subtracting it, whether it is on the east or west of the standard meridian, respectively.

Converting angular distance  into hourly time:

If I want to change the angular distance (longitude) into the time, I use the following relationships:

•  360 degrees(Angular) = 24 hours. (time)             1 hour(time) = 15 degrees.(angular)
• 1 degrees (angular) = 4 min.  (time)                     1 min (time)= 15 minutes(angular)
• 1 min(angular) = 4 seconds(time)         1 seconds(time) = 15 seconds(angular)    

So a longitude of 82 degrees 30 minutes   = 5 hours 30 minutes

Thanks

What are Mistakes,Systematic error and Accidental Errors, in Surveying

Hi,
This post contains mostly the theory of the Errors, as we define them in Surveying. 

Error:- The difference between the observed value and the true value is known as error.

There are three types of errors which occurs while we do the surveying:

1. Mistakes: These are the errors which occur due to the inexperience, inattention, carelessness or due to lack of judgement or poor judgement. If mistakes are not found then they may affect the result to a great extent.
2. Systematic errors: These are the errors which follow a system when they occur, and they have the same nature whenever they occur. They can be eliminated by testing the instruments before they are used or by applied the necessary correction, by using the mathematical formulae after the error is known.
3. Accidental errors: These are the kind of errors which occur accidentally and can of any nature positive or negative. These are the errors which are byond the human control and can not be calculated to their true value, but only we can apply the theory of the probability to calculate them.

Weight of an observation: Weight of an observation is its relative importance to the other observations taken under the identical conditions.

Mean value: Mean value of a set of observation is the arithmetic or the weighted arithmetic mean of the observations.

Probable Error of a single observation:

Probable error of a single observation in a set of observation is derived using the formula,

Es = +-0.6745√(∑v²/ n-1)

where, v= difference between the observation and the mean value of the set of observation.

And n= numbers of observations in the given set.

Probable error of the mean:

The probable value of the error in a mean of the set of observation can be determined using the following formula = Es/√n

Principle of least squares:

Most probable value of a given quantity from the given available set of observation is the one for which the sum of the squares of the residual errors is a minimum.

Alternatively, the most probable values of the errors in the given set of observations of equal weight are those for which the sum of their squares is a minimum.

It can be proved that the mean value is the true value in case the numbers of observations are very large in numbers.

The sum of the squares of the residuals found by the arithmetic mean value is a minimum. This is thus the fundamental law of least squares. 
Hi,

Adjustments of the errors in the triangulation:
The triangulation errors are adjusted in three different ways or steps:
(a) Adjustments of the single angle error (b) Adjustments of the station observation
(c) Adjustments of the figures


(a) Adjustments of the single angle error: When an angle is measured a numbers of times, then the most probable value of the angle is the mean value of the observed values. If all the values observed are of the same weight then the most probable value is the simple arithmetic mean, but if the observations taken are of different weight then the most probable value is the weighted arithmetic mean.

(b)Station adjustments:
When there are numbers of observations taken at the same station, then the condition that the sum of all the angles should be equal to 360 degrees must be satisfied, if not then the difference from 360 is the error of that station. Now there arise two more cases:
(i) When all the angles measured have the same weight: In this case the error is distributed equally among all the angles. 
(ii) When the angles are of different weight: In this case the error is distributed in the proportion of inverse of the weight of the different angles.  
(iii) When the angles and some other combined observations are taken: In such cases when combined observations are also taken along with the observation of the single angles, we have to make use of the normal equation, to find out the errors. 
There is another method known as the method of difference, which can be used in more simple way to get the errors, because the method of normal equations is more laborious. 

(c) Adjustments of the figures: 
There are different conditions which can be opted to calculate the errors of the different angles. In a triangular figure, the sum of three angles is always equal to 360 degrees. 
Similarly there can be other conditions which can opted to find out the condition equations for other figures like quadrilateral, or central figures. 
For a closed traverse, sum of internal angles is (2n-4)*90 degrees, where n is the no. of sides, or total no. of angles. 

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Relevant Books that you might buy:

       

Trigonometric Leveling - Theory and Formulas for Calculations.

Hi,

Trigonometric Leveling is the branch of Surveying in which we find out the vertical distance between two points with the help of some measurements of the vertical angles and the known distances. The known distances are either assumed to be horizontal or the geodetic lengths at the mean sea level(MSL). The distances are measured directly(as in the plane surveying) or they are computed as in the geodetic surveying.

The trigonometric Leveling can be done in two ways:

1. Observations taken for the height and distances
2. Geodetic Observations.

In the first, we can measure the horizontal distance between the given points if it is accessible.

We take the observation of the vertical angles and then compute the distances using them. If the distances are large enough then we have to provide the correction for the curvature and refraction and that we provide to the linearly to the distances that we have computed. 

In the second, i.e geodetic observations, the distances between the two points are geodetic distances and the principles of the plane surveying are not applicable here. The corrections for the curvature and refraction are applied directly to the angles directly.

Now we will discuss the various cases to find out the difference in elevation between the two Points.

(1) The two points are at known distance, i.e. The base of the object is accessible.

When the two points are at a known horizontal distance then we can find out the distance between them by taking the vertical angle observations.

If the vertical angle of elevation from the point to be observed to the instrument axis is known we can calculate the vertical distance using trigonometry.

Horizontal distance*tangent(vertical angle) = Vertical difference between the two.

If the points are at small distance apart then there is no need to apply the correction for the curvature and refraction else you can apply the correction as given below:

C= 0.06728D*D

Where D is the horizontal distance between the given two points in Kilometers.

but the Correction is in meters (m).

(2) The base of the object is not accessible :

(a) When the instrument is shifted to the nearby place and the observations are taken from the same level of the line of sight: 

In such case we have to take the two angular observations of the vertical angles. The instrument is shifted to a nearby place of known distance, and then with the known distance between these two and the angular observations from these two stations, we can find the vertical difference in distance between the line of sight of the instrument and the top point of the object.

(b) When the line of sights of the two instrument setting is different :

Here again there are two cases:

1.  When the line of sights are at a small vertical distance which can be measured through the vertical staff readings. 
2. When the difference is larger than the staff height.

(i) In first case, It is advised to apply the formula for the difference in the height of the top of the object from these two lines of sights. The difference in lines of sights is same as the staff readings difference, when the staff is kept at a little distance from these two points. So we can get the solution for the vertical distance easily.

(ii) In the second case, there is a need to put a vane staff at the first instrument station and the angle of elevation is measured from the second point of observation. This gives us the difference in the line of the sights between the two points of instrument station. Then again we do the same.

(c) When the instrument station and the top of object are not in same vertical plane:

In this case there is a need to measure at-least two horizontal angles of the horizontal triangle formed by the two instrument stations and the base of the object.

Again we will take the vertical angular observations from the two instrument stations also and then we can apply the sine rule to solve the horizontal distances of the triangle. With the help of these angles and the distances we can get the vertical distance between any two point(Instrument station and the top of object).

Reference:

       

Thanks!

What is Local Attraction, and How Does it Affect the Readings of Compass.

Hi,

Local Attraction: 

A compass shows the direction of the magnetic meridian on the principle of magnetism. 
Now whenever you bring any magnet attracting material(Ferrous metals), needle will show deflection. Please note that in its undisturbed condition, needle is always pointed towards magnetic north. 

So when that ferrous material is near to it, needle will not longer be in north south alignment, so that is how the error is introduced into your readings. This error is known as the local attraction.

Materials which are most likely to be present there, while you are doing the compass surveying, are such as an iron chain, metallic wrist band or ear rings(metallic) that one might be wearing.

Other things such as an electric pole or electric wires may also produce local attraction. The needle is attracted to these objects, so this will deviate from the true direction of the magnetic meridian.

If local attraction is available at a station then all the readings taken from that station will have the same amount of the error, and we have to correct the readings to get the true results.

There are methods to get the corrections to be applied on the erroneous readings in the traversing. The two methods which are used in general will be discussed here briefly.

(1) In first method we have to find out the stations where no local attraction exists. To find out this we have to look for a line where the difference between the fore bearing and the back bearing is exactly equal to 180 degrees. If we find such line then that means the two end stations of that line are free from any local attraction. After finding that line we apply the correction to the bearings of the other lines.

(2) In the second method we find the line where there is no local attraction. We know that even if the local attraction is present at every station the measured included angles will not be incorrect and we can calculate them correctly. With the help of the  readings from the stations which are free from local attraction and the correct included angles we can find out the bearings of all the lines.

If we do not find any line where the both stations are free from the local attraction, we have to take the line where the error is minimum and then apply the mean correction to both the stations and then take them as the correct readings. After that start as usual.

If you want further assistance with the topic, please leave a comment.

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