Can we make huge scales to measure huge distances? Of course not! Then how do we measure large distances like the distance between the Earth and the Sun? We use our mathematical knowledge about triangles called trigonometry (Trigonometry deals with relationships between the lengths of sides and angles of a triangle) to find such large distances.
With the help of the parallax method (which is a special case of triangulation) the distance of far away objects can be measured or estimated. The parallax method has been used throughout the years by astronomers to map the 3D world by observing the 2D sky from the earth.
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It is a process of finding out the location of an object/point in space by forming a triangle to the point from given/known points.
When you hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye O_{1} (closing the right eye) and then look at the pencil through your right eye O_{2} (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax.
The distance between the two points of observation is the basis for calculation of other lengths, b (O_{1}O_{2}).
Here two points of sight or observers are at a distance, b from each other. They are observing a distant object P. P is very much far from them, so that O_{1}O_{2 }can be considered as an arc of radius s subtending angle ϕ (in radian) at P.
So b=s ϕ
We know the distance between these two points of sight i.e. b and ϕ is measured. From the above equation, we can get the distance, s.
This method is widely used in determining the distance of faraway objects (e.g. stars, planets, ships at the horizon etc).
Let O_{1} and O_{2} are two points of sight on the earth and we are observing a planet P. There is also a star, S far away from the earth which can be treated at same angle of incidence to O_{1} and O_{2} i.e. just above head, at 90°.
The planet is observed to be at angles ϕ_{1} and ϕ_{2 }from O_{1} and O_{2} respectively.
The points of sight O_{1} and O_{2 }are at distance b. The planet is quite far from the earth and can be considered as if it subtends an angle 1+2 with radius s to the points of observations ϕ_{1}and ϕ_{2}.
So b=s (ϕ_{1}+ϕ_{2})
We know the distance between these two points of sight i.e. b and the parallactic angle, ϕ=ϕ_{1}+ϕ_{2} (in radian) is measured. From the above equation, we can get the distance, s.
From the procedure explained before we can measure the distance of a near planet, s.
Now, from an observatory point, O we measure the angle subtended by the planet i.e. ϕ.
So, P_{1}P_{2}=d where d is the diameter of the planet.
So, d=sϕ
Thus, we can measure the diameter of a nearby planet using the parallax method.
For measuring the distance of nearby star S_{1}, we have to take a faraway star S_{2} as reference whose angle of incidence on earth can be treated as constant over the year i.e. above head or 90°.
We are to measure the distance, d of the nearby star from the sun. For that, we measure the angle made by S_{1} with respect to S_{2} at two extremes of earth orbit. Let those be ϕ_{1}and ϕ_{2} respectively. We know the diameter of the earth’s orbit, D.
As S_{1}is quite far from the sun, the diameter of the earth’s orbit can be taken as an arc with an angle ϕ_{1}+ϕ_{2} subtended at S_{1}. So, the distance d can be measured as,
D=d(ϕ_{1}+ϕ_{2})
1 LY= 9.461×10^{15} m
Parsec is a unit of length which is used to measure large distances mainly for astronomical objects outside the solar system. It is approximately equal to 3.26 LY or 206×103AU. It is defined as the distance at which one astronomical unit subtends an angle of one arc second. The nearest star Proxima Centauri is about 1.3 pc.
Question.1 From a space observatory satellite near earth, the planet Saturn subtends an visual angle of 77 10^{-3} radian. From doppler test it is found that the distance from the satellite to Saturn is 1.5107 10^{9} km. What is the diameter of Saturn?
Answer. Here, d=sϕ
Question.2 The sun’s angular diameter is measured to be 1920''. The distance of the sun from the earth is 1.496×1011 m. What is the diameter of the sun?
Answer.
So, d=sϕ=13.92×10^{8} m
Question.3 The moon is observed from points A and B on the earth, A and B are diametrically opposite to each other. The angle subtended at the moon by the two directions of observation is 1°54' .Given the diameter of the earth to be about 1.276×107 m ,compute the distance of the moon from the earth.
Answer. Here,
Question.4 A man observes a speed boat at the horizon (5 km away). The speed boat covers an angular displacement of 0.04 radian in 10 s. What is the speed of that speed-boat ?
Answer.
The speed boat covers 200 m in 10 s.
So the speed will be 200 m / 10 s = 20 m/s
Question.1 Measuring distance of nearby star by parallax method is accurate when the distance of the star is
Answer. less than 100 LY
Question.2 Parsec is the distance which subtends an angle of
Answer. 1 second.
Question.3 Earth’s orbit around the sun has radius nearly,
Answer. 1 AU
Question.4 Very large distance can be measured by,
Answer. Parallax method