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Measurement of Large Distances

The following indirect methods are used for measuring large distances:
  1. Reflection or Echo Method
Suppose we wish to measure the distance of an inaccessible wall or a hill from a certain point on the surface of the earth. Standing at that point we fire a gun by directing it towards the wall and we measure the time interval t, between the instant the gun is fired and the instant the echo is heard.

If ‘t’ is the time taken by sound to travel from the observer to the wall and back to the observer, i.e. t is the time taken by sound to travel a distance 2S, where S is the distance between the observer and the wall.

If u is the speed of sound, the distance S,

This principle is used in radio detection and ranging (RADAR) which can be used for determining the distance of a distant object, such as an aircraft.

Radio waves are sent upwards towards the aircraft from a transmitter on earth. The radio waves reflected from the aircraft are received back on the earth and detected by a detector. If t is the time interval between the sending and receiving of the radio waves, the distance S of the aircraft from the earth is computed from the relation
where c =3.0 × 108 ms–1 is the speed of radio waves in air.

The SONAR (SOund NAvigation and Ranging) also works on the same principle. The SONAR is used for the detection of submerged rocks or submarines and for finding their depth. Ultrasonic Waves, which are inaudible sound waves of frequency greater then 20,000 Hz, are sent from a transmitter downwards into the sea.

The waves reflected from a submarine or a rock is detected by a detector. Knowing the speed of sound in seawater and the time interval between the sending and receiving of the ultrasonic, the depth of the submarine or rock can be determined.

The distance of the moon from the earth has been measured to a very high degree of accuracy using a LASER beam. The term LASER is an abbreviation of ‘Light Amplification by Stimulated Emission of Radiation’. Laser is a remarkable device that produces a highly monochromatic, intense and parallel beam of light that can travel very long distances without appreciable deviation.

A laser beam is sent to the moon and the reflected beam is received on the earth. Knowing the speed of light and the time interval between the sending and receiving of the beam, the distance of the moon from the earth can be determined, the error in the measurement being only a few centimetres.
  1. Triangulation Method
Astronomical distances are measured by the triangulation method, which uses the geometrical properties of a triangle in a plane. The principle of this method is as follows: Consider a triangle SAB [Fig.2.1 (a)]. If the length AB of the base and the angles θ1 and θ2 subtended by the point S at the base AB are known, then, using trigonometric relations, the vertical distance h of S from the base line AB can be determined.

The Triangulation Method

Suppose S is a satellite and A and B are two points on the surface of the earth. Then the height of the satellite can be determined by measuring angles θ1 and θ2 by pointing a telescope toward the satellite standing at points A and B respectively.

Using this method, the height of the first Sputnik was found to be 500 km.

This method has to be modified if the point S (e.g. a star) is too far away from the earth, for in that case, θ 1 = θ 2 =  = θ 2 =  900 (and theheightwill come out to be infinity) unless the points A and B are very far away.

The length of the base line can be increased by pointing a telescope at the star at a six-month interval when the earth has moved from one point of its orbit to a diametrically opposite point. In the figure above, these points are A and B. The length of the base line is now increased and is equal to the diameter of the earth’s orbit round the Sun (~3 × 108 km).

Let S be the nearby star whose distance is to be measured. We choose, now, a very distant star (say N), whose direction may be taken to be practically the same at all positions of the earth in its orbital motion.

Suppose at some position of the earth (22 December), say A, we measure, with the help of a telescope, the angle between the direction of the distant star N and that of the star S, i.e. the angle between AN and AS.

Then we wait for six months and measure the angle again when the earth is, say, at position B (22 June).

The sum of the two measured values θ1 and θ2 is equal to θ, the angle the star subtends on the earth’s orbital diameter, D = AB. The angle θ is called the parallax of the star.

Thus θ= θ1+ θ2
In triangles ASC and BSC, we have

For a distant star θ1 and θ2 are small and we can set tan θ1 = θ1 and tanφ2 = φ2 where φ1 and φ2 are in radians. Therefore, we have



Thus, knowing the diameter D of earth’s orbit around the sun and the parallax θ (in radians) of the star, the distance S of the star from the sun can be determined.

For very distant stars (such as N), the triangulation method cannot be used since then θ=0 and cannot be measured accurately. Distances of such stars are measured by spectroscopic methods, about which you will learn in higher classes.

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