There are large no of ways by which we can measure the distances of the heavenly bodies.Let’s find the distance of the planets from the earth’ surface. Let’s discuss the Parallax Method.

**a. Parallax Method:** – Suppose a planet P is seen from the two distinct points present on the earth’s surface. Lets say A and B. Calculate the magnitude of the Parallax angle i.e. the angle formed at point B using the two distances AP and BP as shown in figure below:

The line AB can be called as basis. D is the distance from the points present on the surface of earth to the planet. So

=arc (AB) / radius (PA)

=b / D

D= b /

If in the above equation Electrostatic figure 4.2 will be in radians, and the value of b will be one astronomical unit (1 A.U). Then the distance between the earth and the planet will come in Astronomical units. Now you think what is Astronomical unit?

**Astronomical unit is the average distance of the sun from the earth.
i.e. 1 A.U. = 1.496 X 10 ^{11}m.**

Other method to calculate the distance of the planet from earth is Copernicus method.

**2. Copernicus Method: **- This method is basically used to find the distance between those planets which are near to sun rather than that of earth. Before applying this condition we have to suppose that the orbits in which the planets are moving are circular. To calculate the proper distances, using the observations we have to find the angles between both the directions i.e. (earth to planet and earth to sun). The angle so formed is called as the angle of planet’s elongation.

Let’s talk about the planet P and our earth E. Both are revolving in particular orbits around the sun having radius r1 and r. This is shown in the figure below:

In the above figure r1 is the distance of the planet from the sun and r2 is the distance of the planet from earth.

The revolving of planets around the sun is a continuous process. The distance during revolution i.e. r remains same throughout. But the distances that vary are r1 and r2. Along the distances the angle PES also goes on changing. When the angle formed by sun and earth over the planet will be equal to 90^{ o }then the elongation of the planet will reach its maximum value.

Now consider the triangle SPE present in the above figure. So

PS / SE =Sin

Or we can write it as

PS=SE Sin

Because r1=r Sin Electrostatic —— 4.1

Also

PE / SE =Cos

Or

PE =SE Cos

Then r2=r Cos ——- 4.2

A method known as radar estimation method can be used to find the approx distance between the sun and the earth. According to this method

1A.U. = 1.496 X 10 ^{11}

To find the distance r2 a common method is used. We will discuss it in detail.

To find r2 a radar transmitter is used which generates radio waves. Then these radio waves are caught with the help of a radar receiver. Then the time of echo is also recorded.

Suppose the time taken by the radio waves to reach the planet and coming back is denoted by t. Then the total distance of the planet from the earth will be :

r_{2}=ct / 2,

In the above equation c is the velocity of the light.

Now substitute the values of r and r2 that were calculated above in the equation 4.2. After substituting and solving the equations the value of can be calculated. Then further the value of will help us to find the value of r1 from equation 4.1

3. Third method to find the distances between the heavenly bodies is the **Kepler’s Third Law.** Kepler’s law can be applied to those bodies whose orbital track is larger than the earth’s orbit around the sun.

According to Kepler’s law

T^{2} R ^{3}

In the above equation R represents the semi major axis and the T is the time period.

Suppose R _{1} and R _{2} are the semi major radii and T _{1} and T _{2} are the time periods then

T_{1}^{2} / T_{2}^{2} = R_{1}^{3} / R _{2}^{3}

So

R_{2} = R_{1} (T_{2}/ T_{1}) ^{2/3}

If we have the values of R_{1}, T_{1} and T_{2} then we can easily find the value of R_{2}.

4. Fourth method to find the distance between the heavenly bodies is **Spectroscopic Method.** To calculate the distance of stars which are at a larger distance Spectroscopic Method is used. The data needed to perform calculation by this method is the intensities of both the bodies whose distances have to be found. Suppose r2 and r1 are the distances of the star that are near to earth and the distance of star that is at a larger distance from earth. We know that the square of the distance between the stars is inversely proportional to their intensities.

I_{1} / I_{2} = r_{2}^{2} / r_{1}^{2} ——- 4.1

If we have the value of the r_{2} i.e. distance of the nearby star, then by using the value of r_{2} we can calculate r_{1} very easily.