Question

Earth is 1.495 × 10⁸ km from the sun, and another planet is 5.896 × 109 km from the sun. If Earth has an orbital period of 365 days, what is the orbital period of the other planet in Earth days?
1. 32440 days
2. 90400 days
3. 50000 days
4. 11232 days

Answer 1

Correct Answer - Option 2 : 90400 days

The correct answer is option 2) i.e. 90400 days

CONCEPT:

• Kepler's laws of planetary motion
  • The first law (Law of orbits): All the planets revolve around the sun in elliptical orbits having the sun at one of the foci.
  • The second law (Law of areas): The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time.
  • Third law (Law of periods): The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis.

​CALCULATION:

From the third law of Kepler, the square of the time period (T) is directly proportional to the cube of the semi-major axis (R).

⇒ T2 ∝ R3

Given that:

Time period of Earth, T_e = 365 earth days

Mean distance of Earth from the Sun, Re = 1.495 × 108 km

Mean distance of the planet from the Sun, Rp = 5.896 × 109 km

If T_p is the time period of the other planet in Earth days,

\((\frac{T_e}{T_p})^2 = (\frac{R_e}{R_p})^3\)

\(⇒(\frac{365}{T_p})^2 = (\frac{1.495 \times 10^8}{5.896 \times 10 ^9})^3\)

⇒ Time period of the other planet, T_p = 90400 earth days