Question

The semi-major axis of the orbit of Saturn is nine times that of Earth. The time period of the revolution of Saturn is approximately equal to?
(A) \[81years\]
(B) $27years$
(C) $729years$
(D) $\sqrt[3]{{81}}years$
(E) $9years$

Answer 1

Hint:-From the question we know that the semi-major axis of Saturn is nine times that of Earth's. For a relationship between the time period and the semi-major axis, we use Kepler’s law of time periods. Kepler’s law states that a planet moves in such a way that the square of its time period is directly proportional to the cube of the semi-major axis of its orbit. The mathematical representation for this is given as,${T^3} \propto {r^3}$ here, $T,r$are the Time period of the planet and length of the semi-major axis respectively. For solving the problem we divide the equation of Kepler’s law for Earth and Saturn to get the time period of Saturn. Units of the time period are taken as Earth years usually and major axis units as kilometers.

Step by Step Solution:
In the question we are given ${r_s} = 9{r_e}$, where ${r_s},{r_e}$ are semi-major axes of Earth and Saturn. Taking ${T_s},{T_e}$as time periods of Saturn and Earth respectively, now using kepler’s law for both the planets and dividing them,

${T_e}^2 \propto {r_e}^3.........(1)$
${T_s}^2 \propto {r_s}^3 $
$ {T_s}^2 \propto {(9{r_e})^3}........(2)$
Dividing (2) with (1) we get,
$ \dfrac{{{T_S}^2}}{{{T_e}^2}} = {(\dfrac{{9{r_e}}}{{{r_e}}})^3} $
$ {T_S}^2 = {9^3} \times {T_e}^2 $
${T_S} = \sqrt {{9^3} \times {T_e}^2}$
${T_s} = {9^{\dfrac{3}{2}}} \times 1year$
We take the time period of Earth as one year because from the definition of the time period it takes one year for planet Earth to complete one revolution.
${T_s} = 27years$

Hence option (B) $27years$ is the correct answer.

Note:This method is not only used for planets but also used when we have to deal with satellites. Kepler was the scientist that proved that the sun is the center of the orbit using mathematical data as evidence. Although it was proposed by Copernicus. Kelper made three laws, they are

1) The path of the planets about the sun is elliptical, with the center of the sun being located at one focus. (The Law of Ellipses)

2) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)

3) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)