One year at a planet is 8 times as large as compared to the one year at the Earth. Which one of the following is correct about the planet's orbit?
- A.The semimajor axis of the planet's orbit is same as that of the Earth.
- B.The semimajor axis of the planet's orbit is twice as compared to that of the Earth.
- C.The semimajor axis of the planet's orbit is thrice as compared to that of the Earth.
- D.The semimajor axis of the planet's orbit is four times as compared to that of the Earth.
Correct answer
Explanation
The correct answer is D, four times that of the Earth. Kepler's third law says the square of a planet's orbital period is proportional to the cube of the semimajor axis of its orbit.
Writing it as T² ∝ a³, a period 8 times the Earth's gives T² = 8² = 64 times. The cube of the semimajor axis must then be 64 times, and the cube root of 64 is 4. So the planet's average distance from the Sun is four times the Earth's, about 4 astronomical units. Kepler published this law of periods in 1619; Newton later showed that it follows from the inverse-square law of gravitation.
A, the same axis, would give the same year as the Earth, not one eight times longer.
B, twice the axis, gives a period of √8, about 2.8 Earth years.
C, thrice the axis, gives √27, about 5.2 Earth years, still short of 8.
Exam tip: T² ∝ a³; a year 8 times longer means 4 times the distance, and 27 times longer means 9 times the distance.
Practice Questions
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