Sunday, March 24, 2013

Field Astronomy - Zenith and Nadir


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 = 24 hours. (time)             1 hour(time) = 15 degrees.(angular)
1 degrees = 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


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Tuesday, March 12, 2013

Errors in Surveying - Mistake,Systematic error and Accidental Errors


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. 

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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