Imagine a planet that orbited the sun at a distance three times that of Earth. What would the period of that planet be in Earth years?
To determine the period of a planet orbiting the sun at a distance three times that of Earth, we can use Kepler's Third Law of Planetary Motion. According to this law, the square of the period of a planet's orbit is directly proportional to the cube of its average distance from the sun.
First, we need to find the ratio of distances. The average distance of Earth from the sun is approximately 149.6 million kilometers. If the new planet orbits at a distance three times that of Earth, then its average distance would be 3 * 149.6 million kilometers.
Next, let's calculate the period ratio using the distances. The ratio of periods of two planets can be found by taking the square root of the distance ratio cubed. Let's denote the period of Earth as Te and the period of the new planet as Tp:
(Tp / Te) = (dP / dE)^(3/2)
Now, substitute the values:
(Tp / 1 year) = [(3 * 149.6 million km) / (149.6 million km)]^(3/2)
Simplifying:
(Tp / 1 year) = 3^(3/2)
Calculating:
(Tp / 1 year) ≈ 5.196152
Therefore, the period of the new planet would be approximately 5.196152 Earth years.