Question

The time taken by the planet to revolve around the sun is 1.5 earth years. Find out the ratio of semi major axis of planet's orbit to semi major axis of earth's orbit around sun.

Answer 1

Kepler's law of planetary motion:

The third law or Law of periods states that the “ Square of the time period of revolution of the planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis ”

Mathematically, $T^2\alpha a^3\to \left(1\right)$

where T is the time period of revolution and a is the length of the semi-major axis of the elliptical orbit.

Given data:

The time period of the planet, $T_P=1.5\text{yrs}$

We know, the time period of earth $T=1\text{yr}$

Let $a_P$ be the semi-major axis of the planet and $a$is the semi-major axis of the earth.

Using relation (1), we can compare the time periods of the planet and earth as

$$\begin{gathered} \frac{{T_P}^2}{T^2}=\frac{{a_P}^3}{a^3} \\ \frac{a_P}{a}={\left(\frac{T_P}{T}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} \end{gathered}$$

Therefore,

$\begin{array}{rcl}\frac{a_P}{a}& =& {\left(\frac{1.5}{1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ & =& {\left(1.5\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ & =& 1.31\end{array}$

Thus, the ratio of the semi-major axis of the planet's orbit to the semi-major axis of the earth's orbit around the sun is ${\left(1.5\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=1.31$