Question

The time period of earth is taken as T and its distance from sun as R. What will be the distance of a certain planet from the sun whose time period is 64 times that of earth?

Answer 1

Use the law of time periods of Kepler's law of planetary motion which states that “the planet moves in such a way that the square of its time period is directly proportional to the cube of the major axis of its elliptical orbit”. Mathematically,
${T^2} \propto {R^3}$
Where, T = Time period of the planet.
R = Length of semi major axis.

Complete step by step solution
Given: T = Time period of Earth.
R = Distance of Earth from Sun.
64T = Time period of the planet.
Let the distance of the planet from the Sun be RP.
From Kepler’s law of planetary motion we know that,
${T^2} \propto {R^3}$……(i)
For the planet the same equation becomes
${(64T)^2} \propto R_P^3$……(ii)
Now dividing equation (i) by equation (ii) we get
$\dfrac{{{T^2}}}{{{{(64T)}^2}}} = \dfrac{{{R^3}}}{{R_P^3}}$
Or, $R_P^3 = {(64)^2}{R^3}$
Or, $R_P^3 = {({4^2})^3}{R^3}$
Taking cube root on both sides we get
Or, ${R_P} = 16R$.

Note This method of solution can also be used when we are dealing with satellites. It should be noted that it was Copernicus who first introduced the idea that Sun is the central body in our planetary system; later Kepler confirmed it by putting it on a solid mathematical background. Although Kepler’s laws aren’t really lawful in the way we commonly think as the orbital path of the planets around the Sun are not perfectly elliptical.
The three laws of Kepler are as follows:
(i) Law of elliptical orbits: The planets revolve around the Sun in an elliptical orbit with the Sun as one of its foci.
(ii) Law of areal velocities: The planets revolve around the Sun in such a way that its areal velocity is constant or line joining the planet and the Sun sweeps out equal area in equal interval of time.
(iii) Law of time periods: Also known as the harmonic law. It states that the 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 elliptical orbit.