Question

At a distance of 1 A.U., the earth takes one year to travel around the sun. How far away is a planet that takes eight years to go around the sun?

Answer 1

Answer:

4 AU

Explanation:

We can solve the problem by using Kepler's third law, which states that the ratio between the square of the period of revolution of a planet around the Sun and the cube of its average distance from the Sun is constant for every planet orbiting the Sun:

$\frac{T^2}{r^3}=k$

where

T is the orbital period

r is the average distance of the planet from the Sun

If we take two planets 1 and 2, the equation can be rewritten as

$\frac{T_1^2}{r_1^3}=\frac{T_2^2}{r_2^3}$

In this problem, we have:

$T_1 = 1 y$ is the orbital period of the Earth

$r_1 = 1 AU$ is the distance of the Earth from the Sun

$T_2 = 8 y$ is the orbital period of the second planet

Therefore, we can re-arrange the equation to calculate r2, the averag distance of the other planet from the Sun:

$\frac{r_2^3}{T_2^2}=\frac{r_1^3}{T_1^2}\\r_2 = \sqrt[3]{\frac{r_1 ^2 T_2^2}{T_1^2}}=\sqrt[3]{\frac{(1 AU)^3(8 y)^2}{(1 y)^2}} =4 AU$