Question

A planet orbits the sun with a period of 14 years. How far is the planet from the sun?

Answer 1

Using Kepler's Third Law of Planetary Motion, which relates the orbital period of a planet to its distance from the Sun, a planet that orbits the Sun with a period of 14 years is approximately 5.8 astronomical units (AU) away from the Sun.

Step-by-step explanation:

To determine the distance of a planet from the sun given its orbital period, we can use Kepler's Third Law of Planetary Motion. This law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit (the average distance from the Sun). In this case, the student is asking about a planet that has an orbital period of 14 years.

Specifically, for our solar system, if we express the period in Earth years and the distance in astronomical units (AU), Kepler's Law can be simplified to p^2 = a^3, where 'p' represents the orbital period in Earth years, and 'a' represents the semi-major axis in astronomical units. For Earth, p is 1 year, and a is 1 AU, by definition. To find the average distance 'a' for the planet in question, we solve this equation for 'a'.

Let's calculate:

1. p = 14 years (given orbital period)
2. a^3 = p^2 (according to Kepler's Law)
3. a^3 = 14^2
4. a^3 = 196
5. a = ∛196
6. a ≈ 5.8 AU

Therefore, the planet is approximately 5.8 astronomical units away from the Sun.