Question

Explain with a diagram how the distance of a far away planet from earth can be calculated using parallax method.

Answer 1

We are asked to calculate the distance of a planet far away from earth using the parallax method. Parallax can be defined as an effect in which the position or direction of an object is obtained to be different when viewed from different positions.

Formula used:
The formula of parallax method is,
$D=\dfrac{b}{\theta }$
Where, $D$ is the distance to the star or planet, $\theta$ is the parallax angle to the star or planet and $b$ is the distance between the points on earth from where the star or planet is observed.

Complete step by step answer:
If we see an object closing our left eye then see it closing our right eye, we will observe a shift or change in the position of the object. This shift in position is called parallax. The distance of a planet or star far away from earth is measured using the method of parallax.

Let the distance of the planet $\text{P}$ from the earth is $D$ and let us assume that the distance of the planet $\text{P}$ is measured from two different positions on earth that is point $X$ and $Y$ at the same time.First we draw a schematic diagram for the problem.

The distance between points $X$ and $Y$ is $b$ and the angle $\theta$ is the parallax angle. The distance between the earth and the planet is very large so we can assume that, $\dfrac{b}{D}<<1$ and $\theta$ is a very small angle. Therefore, we can approximate $XY$ as an arc which has a length $b$ of a circle whose centre is point $\text{P}$ and we can take the radius of the circle as $D$.

The length of an arc can be written as the product of its angle with the centre and radius of the circle. So, using this concept the arc length $b$ can be written as,
$b=\theta D$
$\Rightarrow D=\dfrac{b}{\theta }$
Therefore, if we know the values of the shift in position of the two observing positions on earth and the parallax angle then we can calculate the value of the distance of a far away planet from the earth using the formula,
$\therefore D=\dfrac{b}{\theta }$

Note: In such types of questions, always draw a diagram before proceeding for calculations as it will help you to visualize and understand the problem well. Also, after calculating the value of the distance of the planet from the earth we can calculate the angular diameter or size of the planet using the method of parallax.